25
INFORMATIONAL LOGIC III
INFORMATICA 1/89
Descriptors: LOGIC INFORMATIONAL, RULES FORMATIONAL, AXIOMS INFORMATIONAL, INFORMATIONAL WELL-FORMED FORMULA (IWFF)
Anton P. Železnikar Iskra Delta, Ljubljana
In this part of the essay two main topics of tha informational logic (IL) are discussed: formation rules which govern the structure of informationally well-foraed formulae (iwffs) and infornational axioms. In the continuation of this essay informational transformation rules of IL will be examined in a formal informational way.
Formation rules of IL have to answer the question how to construct initial informational formulae which will belong to the so-called class of iwffs. Within this question the so-called operational, operand, and parenthetic constituents and their compositions into iwffs are determined. In formatting a formula (iwff) several basic processes can be applied, for instance, beginning of formula formation, introducing of operands and operators in implicit and explicit forms into the context of an iwff, particularization and universal ization of formulae, etc. Afterwards, formation rules of IL are exposed in a short and concise manner. At the end, the question can be put what could be the form of a non-informational formula.
Within the topic of informational axioms the following subjects are discussed: axiomatization of informational principles, how informational axioms can be generated, axioms of Informing! informational difference, informational circularity, Informational spontaneity, informational arising, counter-inforoation, counter-Informing, informational embedding, informational embedding of counter-information, informational differentiation, informational integration, informational particularization and universalization, informational structure and organization, informational parallelism, informational cyclicity, openness of informational axiomatization, influence of metaphysical beliefs on axiomatization., and axiomatic consequences of informational arising.
Informacijska logika III. V tem delu spisa se obravnavata dve glavni naslovni poglavji informacijske logike (IL): formacijska (oblikovalna) pravila, ki urejajo Btrukturo informacijsko dobro oblikovanih formul (iwff) in informacijski aksiomi. V nadaljevanju tega spisa se bodo na informacijsko formalen naC i n preučevala Se informacijska transformacijska pravila IL.
Formacijska pravila IL morajo odgovoriti na vpr&Sanje, katere informacijske formule pripadajo t. i. razredu iwff. V okviru tega vprašanja se opredeljujejo operacijski, operandni in oklepajni konstituenti in njihove kompozicije v iwff. Pri oblikovanju informacijskih formul (iwff) se uporabljajo nekateri osnovni procesi, kot so npr. začenjanje oblikovanja formul, uvajanje operandov in operatorjev v implicitni in eksplicitni obliki v kontekst formul, atikanje operatorjev, partikulariziranje in univerzalieiranje formul itd. Oblikovalna pravila IL je mogoCe opisati kratko in jedrnato. Vprašanje, ki ga je mogoCe postaviti pri oblikovanju formul je, kakSna bi lahko bila oblika neinformacijske formule.
V okviru problematike informacijskih aksiomov pa se obravnavajo tale naslovna vpraSanja: aksiomatizacija t.i. informacijskih principov, kako je mogoCe generirati aksiome, aksiomi informiranja, informacijske diference, informacijske cirkularnosti, informacijske spontanosti, nadalje aksiomi informacijskega nastajanja, prot i informacije , protiinformiranja, informacijskega vmeSCanja, informacijskega vmeBCanja pro ti informac i je, informacijske diferenciacije, informacijske integracije, informacijske partiku1 ar izaci je in uni ver zal izaci j e , informacijske strukture in organizacije, informacijskega paralelizma, informacijske cikliCnosti ter Se odprtost informacijske aksiomatizacije in aksiomatiCne posledice informacijskega nastajanja.
26
II. 2. FORMATION RULES OF INFORMATIONAL LOGIC
... in rejecting mental representations as the objects of belief one is not thereby rejecting the empirical hypothesis that the brain ia an information processor and thus processes in a neural machine language.
Stephen Schiffer [11] 5 II.2.0. Introduction
In thi8 section (II. 2.) we have to say clearly which kind of informational formula will belong to informational logic (IL). Thus, we shall deal with the question how to construct informational formulae which will belong to the legal form of informational formulae. The word legal will have the meaning of well-formed. We have to state precisely what is an expression composed of informational operands, informational operators, and parenthetic delimiters, in such a way that it will represent the so-called informational well-formed formula (iwff).
In the context of formation rules we must consider that an iwff has to be capable of representing any information, most abstract as well as most ordinary life information, simple as well as most complex one. In this respect, it seems Benseful to put the following question: "Is it possible to state explicitly what will be the limits of formula formation or it: it at all possible to set any fixed limits which would disable realization of any general principle of information?" Within this dilemmatic view of formation of an informational formula we shall develop some basic rules of formation, not saying that these rules are the only possible ones.
Already within the principles of information ([4] or [10] respectively) it was shown how informational formulae can be composed on the level of natural language. This experience tella us that informational formulae are in no way limited sequences of informational operands and informational operators in relation to spoken and written language. Relationships within information are objective (operand-characteristic) and subjective (operational) and can be changed from the operand to operational states and vice versa during an informational process. ThuB, a local informational operator oan be viewed as an operational variable in a wider informational observation. Due to thiB phenomenology of informational compositions of operands and operators, iwffs are, in general, not structurally limited in any particular way. Limitations can be determined in cases of particular informational theories, concerning, for instance, formal logical systems as traditional symbolic logic, modal logic, etc., which can be conceived as special projections (particularizations) of informational logic.
The next fact to be explicated ia, that sets of objects of IL are generative, i.e. not limited (determined once and for ever) in advance. Only static theories deal with finite and strictly determined sets of objects and as such are understood to be the most informationally primitive (static) forms of IL. In this respect, the set of formation rules of IL will not be semantically limited and informational operators will be recruited depending on needs, goals, and applications, which arise during an informational process. Similar will hold for informational operands occurring in informational formulae. The principle of informational arising will govern the arising of concrete formation rules (concerning concrete informational operators). However, it will be possible to present the essential framework of the arising of formation rules within IL.
II.2.1. Informational Operands as Constituents of IWFF«
The nature of information is variable in its arising, changing, vanishing, and disappearing. An informational operand is such a sort of variable information. We have the following basic definition concerning an informational operand as iwff:
[Operands]^* :
Informational variable a, defined as informational operand, is iwff. This operand can represent various kinds of information belonging to an informational realm. Thus,
('a is informational operand') £
('a is iwff') ■
In many cases it is reasonable to separate informational entities as variable operands. So, one oan set the following definition:
, ,DF2' [Operands] :
A set of informational variables a, 0, ... , Y, in which a, 0, ... , f are defined as informational operands, is iwff. These variables of the set can represent various kinds of information belonging to given informational domains. It is:
('a, 0, ... , f are informational operands')
$ (*a, 0, ... , T is iwff') ■
In the last definition, the commas can be understood as informational operators, which connect operands into an informational Bet. The sequence a, 0, ... , f could as well be written in the following way:
a |=	0 H	. . . (=	T
comma cooua	"comma 1
where |=	ie the	set-connective
comma
informational operator representing the delimiter ",". A set of operands a, 0, ... , f can be represented by a resultant operand, say 5, where
? = a, 0......
27
In this case, the symbol '=' is the informational operator of representation. It has the meaning that ? representatively Informs a, 3. ... , f or
5 t=_ a. P......
Let [Operands]
us set now the definition reverse to DF1.
[Operands:
If a represents an iwff, then a is an informational operand:
('a is iwff') =>
('a is informational operand') ■
This definition says that irrespective of how a is structured as iwff, in fact, it represents an informational operand (traditional variable). Or, in other words: irrespective of its structure, an iwff can always be used as informational operand or can be put under operation of an. informational operator. Bverything which is iwff can be operated or can become an operand in the structure of another, higher formula. In the iwff
P i
a, p, ... ,7 are iwffs, so, each of these iwffs can have its own structure.
DP 4
[Operands] :
If <x, p, ... , -f are informational operands,
then 5f in SJ(«> P.....T) ia the so-called
functional informational operand (fio) or implicit informational operator (iio). In an iwff, a fio performs as informational operand, however, it haB the implicit operational property. In this respect, an informational operand can be a functional or a non-functional variable. Fios or iios will be marked by capital Oothic letterB (for instance, SI, ©, <E, etc.	■
In JJ(a, P, ... , T>> the operand variables a, P, ... , T can be functional as well as nonfunctional. Some distinguished iios (or fios) are, for instance:
«
S S
5 3
general Informing, for instance, as Informing of the variable a, behavioral Informing or behavior of a being,
counter-Informing of information, informational differentiation (which could be the synonym for counter-Informing) ,
informational embedding (of counter-information or new information into source information),
general implicit functional operator, Informing or informational integration,
motor or behavioral Informing of a being,
metaphysical Informing of a being,
V informational particularization (subscription of informational operators) and informational universalization (superscription of informational operators), 6 sensory Informing of a being, etc.
Marking by Gothio letters does not mean that alBO capital Latin letters cannot be used
DF1
(according to (Variables] ) for the purpose of marking implicit operators within thé operand expressions. In this sense, for the above list of markers of Informing, also the Latin letters A, B, C, D, E, F, I, L, M, P, S, etc. oan be used, respectively. Gothic letters are introduced for better distinctness of implicit operators in the expression of operands.
II.2.2. Informational Operators as Constituents of IWFFs
In IL operators are understood to be variable. In general, informational operators, presented by the metasymbol , will belong to a generative, potentially unlimited set of informational operators. In contrast to the so-called implioit informational operators narked by capital Gothic letters, metaoperators belong to the so-called explicit informational operators, which will be marked by distinctive special symbols. The set of informational operators is generated by the two already mentioned informational procedures, called particularization and universalization of existing metaoperators or already particularized or universalized operators. The process	of	particularization	or
universalization can always begin from the most general informational operator	which as
metasymbol represents the so-called general operational variable. In IL, on the level of informational operations, we regularly have to deal with operational variables rather than with operational constants. Informational operands, as well as informational operators, underlie the so-oalled principle of informational arising.
[Operators
The informational operator ^ is operational variable and is the sub-iwff. This operational variable represents various kinds of informational operators, which can be generated by particularization and universalization of according to the needs, goals, application, and understanding of an informational formula, which is an iwff representing an informational form, process, or phenomenon. In the saae way as does the operator the particularized and universalized operators underlie the philosophy of their further (recurrent) particularization and universalization. Thus,
is operational variable') ^
Ct= is sub-iwff)
('Hpart is operational variable') $
(*|= . is sub-iwff) part
28
< V
where
is operational variable')
( '(=unlv i8 sub-iwff' )
a particularized explicit
N t is part
operator and (=:Unlv js a universalized explicit operator. Further, it is possible to nark
and
'W H
(^univ) = **(>=)
where is the implicit informational operator
t
of particulariz&tion and	the implicit
informational operator of unlversalization. ■
In several cases It is reasonable to concentrate inforaational operators into (regular) functional compositions of operators. In such cases, the following definition can be adopted:
[Operators	;
A set (type) J= of particularized and
universalized operational variables or
informational operators, marked for example as
> N-i > ... , )= , is the basis from which these 1 6	m
elements can be taken to construct the so-
called operational concatenations in the
following way:
(1)	Hcon = 1=^ > where € is operational concatenation (OC);
<2) »=oon 5 (W\J>' Hhere ■"J 6 18 a recursive definition of OC.
If	is an OC, one can write instead of (1)
'con
and (2):
<1>*	€ H * (*=con S ht>
(2)*	( (N; € |=> A ('!=„„ »arks 0C'>) =>
J	con	.
con j
Expression (2)* is a kind of informational modus (a particular form of the so-called modus informationis):
l=j € K Ct=oon denotes OC')
( 'h \=. is OC1 ) conj
The consequence of [Operators]DF43 is that l=con i8 a word belonging to the set
{t=j. Kj.....H,)* \ H
where () denotes the empty set. Hcon i® a
composition of informational operators in an
arbitrary complex (interweaving) way. In a
particular case, )=con can be the linear (usual,
mathematical) composition of operators. The
complex composition means a parallel
(interweaving)	activity of	operators
constituting	. For instance, we shall allow
con
the notation (="* instead of Hi where in 1="* the relation of dominance |= A HI will be assumed. It is evident that among operators, constituting Kcon' additional dependencies (relations) can be determined.
[Operators]DF44:
If ^con rePresents a Bub-iwff, then h=con is an
informational operator (or operational variable):
( 'hcon is sub-iwff') :>
I 'l=con is an operational variable') ■
This definition says that irrespective of how l=con as a sub-iwff is structured, in fact, it
represents a composite (complex) operational variable in which its components (suboperators) are variables too. Formally, as a concatenation of operational variables, t=con functions as an operator composition. In regard to an iwff, a sub-iwff is in some way a reduced form of the iwff concept. This reduction is semantically presented as the prefix 'sub' in sub-iwff, which is a concatenation of informational operators and is marked by operator
II.2.3. Parenthetic Delimiters and Parenthesizing of IWFFs
In fact, parenthetic delimiters can be understood as the delimiting informational operators within an iwff. Their function is to determi ne the so—called iwff's unities within a formula. A unity of an iwff can be used as operand of a higher operator structure. For parenthetic delimiters arbitrary symbols can be introduced, for instance, parentheses, brackets, etc. Besides parentheses, it is possible to introduce the so-called non-substantial delimiter! by which the so-called non-substantial part of an iwff will be marked. So, let us have the following definition:
[Delimiters]0171:
Irrespective of their choice, the parenthetic delimiters occurring in an Iwff unite parts of the iwff or the whole iwff, with the intention to define the unit they delimit to be used for some operation over the unit. Parenthetic delimiters occur always in pairs, consisting of the beginning and the ending delimiter, and can be nested within Other pairs of delimiters. Usually, parenthetic delimiters will be denoted by ' < ' for the beginning and by ' ) ' for the ending delimiter. However, also other kinds of delimiters can be introduced, for instance, the pairs [, ] and (, ], all of which can be understood as particularizations of the delimiter operators	and t=en(j > denoting the
beginning and the ending parenthesis.	■
[Delimiters]®^:
Parenthetic delimiters can be used in pairB in such a way, that they delimit an expression within an iwff, which is either an iwff or a sub-iwff. In this case the usual nesting principle of parenthesizing is valid.	■
29
t_ ,. .. ,DF3 [Delimiters J :
A special, unary delimiter is in fact the symbol, marking the so-called non-sub3tantial or self-comprehensive part of an iwff. This delimiter is composed of three consecutive dots, thus '...'. The three-dot delimiter is a legal symbol of an iwff.	■
[Delimiters]8*1:
Considering the previous three definitions, the legal formulae or iwffs are, for instance:
P,

))
With the last formula we can even determine the positioning of parentheses in an iwff. Evidently, this formula can be rewritten as a(p<T))> where the entities a, 0, and f are understood as unsubstantial parts of the formula in question.	■
II.2.4. Some Basic Processes within the Formation of Formulae
So,far, we have used the following basic processes in the formation of a formula:
(1)	Introducing of informational operands as constituents of an arising formula, where the operand as a variable represents an iwff.
(2)	Setting of informational operands into sets of variables, where distinct variables were separated by commas. Such a set of variables was declared to be the iwff.
(3)	Introducing of explicit informational operators of the type in an arising formula and concatenating them by other informational operators into a sub-iwff (OCs) within an iwff.
(4)	Introducing of explicit informational operators and their operational concatenations (OCs) and concatenating them with operands and their informational sets, thus formatting an iwff.
(5)	Introducing of implicit informational operators of the type gr (functional operands) into the operand parts of an arising formula and concatenating them (functionally) to an parenthesized informational set of operand variables.
(6)	Introducing of explicit and implicit informational operators in a particularized and universalized way of their choice. Hven if operational	particularization	and universalization are the basic formatting principles, they could be understood firBt of all as formula transformation principles (see subsection II. 5).
(7)	Formatting a complete formula (iwff) means to use rules (l.)-{6) in an arising manner. In this respect an instantaneous formula can be always developed by further Bteps proceeding from one formational state to the other in a growing (enlarging) or a vanishing (reducing) manner.
II.2.5. Formation Rules of IL
In the previous definitions of the subsection II.2 we gave the rules for formation of an iwff in the following way: it was said what operands and operators constituting a formula are. Then it was said how operands and operators can be combined or concatenated into a formula. Also, the use of the so-called delimiters, which determine the units or subunits of a formuln, was described. There were not any particular restraints for formula formation. In general, combining of operands, operators, and delimiters in the described way, leads to the formation of a formula. In such formatting processes,	particularization	and
universalization of operators are still possible.
Dpi
[Formatting of formulae] :
An informational well-formed formula (iwff) can
be constructed by the uBe of the definitions
■t_	. -DF1	, ,DF4 r_ . .DF4 2
[Operands] - [Operands] , [Operators]
[Operators, an<i [Delimiters pp 3
[Delimiters]	by concatenating (combining)
operands, operators, and delimiters according to the description in subsection II.2.5. With this procedure production of an iwff is assured.	B
DF2
[Formatting of formulae] :
Informational well-formed formulae, being constructed according to the previous definition, can be composed into the so-called informational system. In this system, distinct, iwffs are separated by commas or/and can appear in different lines. Similarly as a single iwff can be marked by a symbol of operand, the informational system can be marked by a symbol of operand. Informational system performs as iwff.	■
Later on, discussing the transformation rules, we shall show how formulae can be decomposed into systems and how systems of formulae can be composed into formulae under certain circumstances. In this essay we have given several examples, which clarify and illustrate the principles of formula formatting.
II.2.6. What Could Be a Mon-Informational Formula?
The first approach to the topical question could be in questioning what are already informational formulae. The answer to this counter-question is that all mathematical formulae certainly belong to the class of informational formulae. However, informational formulae are not necessarily mathematical, for mathematics does not deal with the creative component of information (Informing, arising, generating, functional changing, etc.). The semantics of informational operators and operands as variable subjects and objects is generative, while mathematically chosen objects are invariably determined.
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The answer to the topical question is that anything we form out of informational operands, operators, and parenthetic delimiters 1b either iwff or sub-iwff. Certainly, there exist some unessential distinctions. For instance, if we take two informational variables, say a and 0, we can form several compositions, such as
a£, a, p, oc(0 ) , < a) P ,
etc. The first case can be interpreted as
informational concatenation of a and	which
is already known as operational concatenation
of the type t . In a similar way it is con
possible to determine the so-called operand concatenation and the concatenation of mixed operand and operator components. This is already performed in the case of iwff formation. The second case concerns the so-called informational set of components a and 0. The third case can, in general, have two substantially different interpretations: a can be an implicit operator and p its argument, or the parentheses '( ' and ')' are used simply as formula delimiters, where a is an explicit informational operator. In the fourth case we have to do with a similar case as in case three, etc. One can feel that any formula expression can have its informational sense, interpretation, understanding, and that it is not possible to say decisively what an informational formula cannot be.
The last statement is evidently only a consequence of the fact that there is not anything which could not be informational. This "all-embracing" principle governs not only the realm of the real information (metaphysics), but also its concept of self-formalization. In this respect, informational logic introduces a free, legal, and effective concept for dealing with cases concerning the generative, creative, or intelligent nature of information. It has to be said that the introduced semantics of IL is initial and has still to be elaborated and developed to the levels, where it would satisfy more properly the needs of a particular arising of information.
II. 3. INFORMATIONAL AXIOMS
... and in the end one is left with the no-theory theory of meaning, the deflationary thought that the question that now define the philosophy of meaning and intentional] t.y all have false presuppositions.
Stephen Schiffer [11] 3
II.3.0. Axiomatization of Informational Principles
We shall now come to the point at which it will be possible to make an authentic self-experience how axiomatization of the so-called principles of information <[41 or 110 ]) opens the abyss of making these principles formalistic, i.e. to give them forms of iwffs. Again, the semantics of operands and operators will take its informational power. In fact, by axiomatization of informational principles, we are entering the abysmal domain of informational phenomenology. If somebody is in doubt whether this is or is not so as we have stated right now, let him simply follow the experiments and intentions of the subsequent informational axiomatization. The nature of the mentioned abyss is that it is practically inexhaustible, that axiomatization as an informational approach can generate an unlimited and unforeseeable set of axioms, where the end cannot be seen at all. It may sound surprisingly that informational axioms, although intuitively deduced from informational principles, can and will illuminate these principles in a refreshing, new, and theoretically elucidated way.
We are now coming to the point where we have to decide what can be an informational axiom. The basic association of ideas is that the so-called informational principles (discussed in 141 and also in [101) have to be axiomati sed. The first impression is that the main difficulty may lie in the principle of informational arising which concerns all informational entities, operands, as well as operators of a formula. Informational arising is the semantic property of operands and operators. Results of applying informational operators is the arisen information, for instance, counter-information. Simultaneously to the arisen information also Informing of information arises or new informational components are coming into existence. Evidently, these basic mechanisms of information have to be formally axiomatized. To enter into the discourse of informational arising a recurrent and informationa 11y interwoven approach is necessary.
II.3.1. The Hain Axiomatic Principles II.3.1.0. How Axioms Can Be Generated
It is simply possible to follow principles of information [4] and make the list of axioms,
31
which will be constructed in the next paragraphs. Thus, we shall have the following axioms concerning the subsequent informational entities: Informing and mutual Informing of information, informational difference, informational circularity, informational spontaneity, arising of information, counter-information, counter-Informing of information, informational embedding, informational embedding of counter-information, informational differentiation, informational integration, informational	particularization and
informational universalization, informational formula, informational structure and informational organization, informational parallelism, informational cyclicity, openness of informational axiomatization, informational axioms and metaphysical beliefs, and axiomatic consequences of informational arising.
(4)	*
((a hP) v (Ma) MPM) v loH P ) )
(5)	(a^ß) *
((a |=P)v(M«)HPM)V(a=| ß>)
(6) ((aMI V (M«D
etc.
(IßM) V <a 4 ß)>
The comment to these axioms is that the informational operators (= and 4 have me taequivalent power and that in mutual Informing of information entities all possible cases of Informing of the involved information can be considered.
II. 3.1.1. Axioms of Informing of Information
II.3.1.2. Axioms of Informational Difference
If we say that a marks an informational entity, i.e. information, then it is supposed that thiB entity has the property of inward (own) and outward (concerning other information) informational development or Informing. For this informational property of Informing the operator variable or general metaoperator was introduced. According to our previous discussion we can propose the following axioms:
[Axioms] :
(1)	('a is information') $ ((a JO V (4 a))
(2)	('a is information') $ ((t= a) V (a 4))
(3)	( (a H v H cc) v (|= a) v (a =|)) *
('3 (a) is coming into existence')
a
(4)	( (a H v (4 a) v a) v (a 4)) *
ll«MI V (a=ja>)	■
The comments to these axioms are the following:
(1)	If a is Information, then a informs in one (J=) or another way (4).
(2)	If a is information, then a is informed in one (t=) or another way (4).
(3)	If a informs and is informed in one or
another way, then Informing of a over a
itself (3 (a)) is coming into existence, a
(4)	If a informs and is informed In one or another way, then a informs itself and/or is informed by itself.
r. • ,DF2 [Axioms]
[of Hutual Informing of Information]
(1)	('a and p are informationally
interwoven' ) => ((a h P) V (P 4 a) V (p t= a) V (a 4 0))
(2)	(a H p) *
((a p) v (p 4 a) v (p (= a) v (a 4 P))
(3)	(p 4 at) *
((a t= P) v (p 4 a) v <p H a) v (a 4 p))
Informational difference concerning two informational items iB the most natural informational property. This fact can be expressed by the following axioms:
DF3.
[Axioms]
(1) (('a iB information') A CP is information'))
(<x * P)
(2)	(UMIVIH«)) * (« * P)
(3)	(('a is datum') A ('p is datum'))
* <hn (a = P))
(4)	(('a is information') A
CP is information')) ^	(a = p))
(5)
(6) (7)
( (a N) v (|= a) )	*	(L 5a(a) )
( <4 a) V (a 4))	*	<V«> -J)
8(a)	=	u(a) a K
In (5 ) -(7), 8(a) denotes the difference arising as Informing of a over itself and =
7t
marks the possible equivalence between informational difference and arising of counter-informfction from a.
For instance, a pure logical axiomatic conclusion would be that
(a * p) =>
(a = p))
From (3) it is understood that only data can be equal. Thus, the equality between two informational items is possible in the realm of information, which represents information as data, that is on the informational1 y static basis. Sooner or later informational equality-remains very unnatural and lifeless informational property.
[Axioms]DF4:
If p is information and if a = p, then a is the marker for p. In this case a is the so-called marking information, which has the nature of information p whose marker it is. Formally,
32
(CP is information') A (a = p)) $
('a is the marker of 0')
II.3.1.3. Axioms of Informational Circularity
[Axioms]
DF5.
(1)	(a |=)		(a °t>
(2)	(a h)	i>	<h a)
(3)	<h a)	*	<a h a)
(4)	<h a)		(a N
(5)	((a h> A (h a))		<h a h)
(6)	(a a)	*	((a H V <h a))
etc. If a informs, then it informs itself (1). If a informs, then it is informed (2). If a is informed, then it is informed by itself (3). If a is informed, then it informs (4). If a informs and if a is informed, then it is informed that it informs (5). If a informs itself or if a is informed by itself, then it informs or it is informed (6). Etc. Evidently, axioms of these types can be generated infinitely.	■
[Axioms]
DF6.
(1) (2)
(3)
(4)
(a	h P)
(a	h P>
<<*	1= P> (a h p>
* IUNAIMII
<e h «)
=> ((« 1= P) H
((a h a) A <0 ^ 0) A
(<«M) h (<»h fS>>>
(Axioms]DF7:
(1) ('a is information') *
((a H V ((= a))
(2)
(3)
(4)
((ah) V (h a))
('3a(a) 1b spontaneous')
< (a h) V ((= a))
((a 3a(a>) v (3a(a) N «>)
<<o |=) V <t= a))
(3 (a) (=3 (a) ) a	a
(5) C3a(a) is Informing') $
(3a.((a M V (|= a)))
etc. If a is information, then a informs or is informed (1). Spontaneity in this axiom is hidden in semantics of the metaoperator h- This fact is expressed in (2). If a informs or is informed, then Informing of a, 3a> is an
implicit functional operator over a, which is informed by a or informs a (3). If a informs or is informed, then Informing of a over a, 3q, informs itself or is informed by itself. If 3a(a) is Informing of a over a, then there exists such an a that (.) a informs or is informed (5). Etc. Obviously, axioms of informational spontaneity can be generated infinitely.	■
II.3.1.5. Axioms of Informational Arising
etc. If a informs p or p is informed by a, then a informs and p is informed (1). If a informs p or p is informed by a, then it is possible that p informs a or a is informed by p. If a informs 0 or p is informed by a, then this Informing informs. If a informs P or p is informed by a, then it is possible that a informs itself, 0 informs itself, and that this Informing informs itself. Ktc. Evidently, axioms of these types can be generated indefinitely.	■
Obviously, it can be understood how information and Informing of information perform circularly. Circularity is the basic informational phenomenology.
II.3.1.4. Axioms of Informational Spontaneity
Synonyms of the adjective "spontaneous" may be unforeseeable, generative, and arising. Informational spontaneity is the property of unforeseeable arising of information and of Informing of information. Spontaneity concerns information as informational operand and Informing as informational operator. Spontaneous means to be capable to arise, to come into existence in a possibly unpredictable way. Spontaneity belongs to the most primitive properties of information. This yields the most simple form of the axioms concerning spontaneity.
The principle of informational arising ia the most basic principle of the theory we call informational logic. This principle hides several other, particular informational principles, for instance, the principles of spontaneity, circularity, Informing, counter-Informing, parallelism, etc. In this respect, this principle has an integrative, originating, and conceptual role in the development of informational theories. The principle that all informational is under the protection and influence of arising, which simultaneously is the synonym for coming into existence, changing and vanishing, guarantees the most possible dynamic nature of information, as it is understood by modern common sense. Of course, the question of new semantic power of informational operators and their operands arises: How can they be determined to surpass the traditional mathematical terminology? How can they be treated to overcome the rationalistic philosophical blockade? How can they exclude, for instance, the principle of truth as the only possible logical means? Several of these efforts have been already presented in the previous text of thiB essay.
[AxiomsJDF8:
(1) ((« h) v (h «)) *
('3a(a) arises') V
('3a<t*) i» coming into existence') v
('3 (a) is generated') v
33
('9Q(a) is changing') V ('3a(a> is vanishing') V
(2) <<3a<a) *=> V (»= 3a(a)>) * ('a arises') V
('a is coaing into existence*) V
('a is generated') V
{'a is changing') v
('a is vanishing') v ...
These two axioms have to be treated as a
uniquei axiomatically interwoven syBtem in
which information a and its Informing 3 (a)
a
are mutually dependent.	■
(2)
v <1= 3(			*		
<(3a<oc)	^ari	a	^ari	3 (a) ) a	V
((3a(a)	^exi	a	^exi	3 (a)) <x	V
(<3a<a)	^gen	a	'""gen	3 (a))	V
((3a(a)	^cha	a	^oha	3a<a)>	V
(<3a<a)	^van	a	^van	3a<a)>	V
In generalt it has to be understood that the process of axiomatization of informational arising can be continued indefinitely.
II.3.1.6. Axioms of Counter-Information
The axiomatic question for the last axiom is where do the arising, coming into existence, generating, changing, vanishing, etc. of information and its Informing originate from. The answer to this question is given by the following axiom:
[Axioms] :
(1)	<(cc 1=) V (|= a)) * (a |= 3a(a>)
(2)	<<3a(a) h=) V 3a(a))) *
<3a(a) I* a))
Without changing essentially the meaning of these axioms, they can be widened in the following manner:
(1)	<(a H v <t= a)) *
(a V°°> V < Va) a))
(2)	<(3a(a) h) V 3a(a))) *
(a (* 3a(a)) V <3a(cc) N a))
In this respect, the consequences of information and its Informing are the sane. ■
IAxiom«J8*1:
Interpretation of the last axioms by means of the previous ones can be given through various particularizations of the metaoperator If we introduce particularizations of |=
for 'arises from or causes the arising of'
t=	for 'comes into existence from or
exl
causes the coming into existence of |=gen for 'is generated or generates' ^cha *or '	or changes'
hvan for 'is vanished or vanishes'
npo
then it is possible to express [Axioms] in the following way:
(1) Hoc H v ()= oc) ) i>
(a	^ari	3a(a)	^ari	a)	V
(a	^exi	v*>	^exi	a)	V
(a	^gen	3 (a) a	(= gen	<x>	V
(a	^cha	3 (a) a	^cha	a)	V
(a	^van	3 (a) a	^van	a)	V
Formation, appearance, or coming into existence of the so-called counter-information is a oonsequence of Informing of information. Counter-information is a result of the informational phenomenology of information. Counter-information arises from information, from information as activity over itself. Appearance of other information, which is not counter-informational, may be called sensory or outward information in respect to the so-called source information or information in question.
[Axioms] :
Let u denote counter-information and let u(a) be counter-information which arises from information a. Let the meaning of operators L and J be 'causes the appearance of or 'comes into existence from', respectively. There is possible to set several axioms, for instance:
(1) ((a h) V a)) * (a L u»(a)) <2> (H a) v (a H>) * <w(a> a) <3) ((a h) V (H a)) *
((a ^ u(a)) A <u<a> ^ a)) (4) (H a) V <oc H>> *
(<u(a) =\ a) A (a w(a>))
etc. The axioms (1) and (2) are already particularized beoause counter-information u(oc) appears as the consequence of information a. In the axioms (3) and (4) the most general operators (= and H are used.	■
We have already shown some axiomatic constructions concerning counter-information in
[Operators]DF^ and [Operators . Evidently, axiomatic constructions concerning counter-information can be continued indefinitely.
II.3.1.7. Axioms of Counter-Informing of Information
Counter-Informing is a component of Informing by which information is producing its counter-information. This component is interwoven in the Informing of information. It acts upon information as a subject causing the appearance of counter-information . Similarly as information is informing, counter-information is counter-informing. The acceptance of counter-information by information depends on the so-called informational embedding of counter-information into the source
34
information. Counter-Informing belongs to the most subtle processing components of information.
[Axioas] :
Let K be counter-Informing and let <£(a) denote counter-Informing which arises within the Informing of information a. It is possible, for example, to set the following axioms:
(1)	(<ct h) V a)) }
( (a L 5(a) ) A (S(a) N <o(a) > )
(2)	(Ha) V (a H) ) *
((«(a) J a) A <w(a) H 1(a)))
(3)	((a |=) V (h= a)) i>
<(a <E(oc)) A (K(a) (= a) A (S(a) k w(«>) A (w(a) (= a))
(4)	<H «) V (a =0) *
( (K( a) =4 a) A (a H «<<*)) A
(u(ot) H «(a)) A <a =4 w(oc)))
etc. In the axioms (1) and (2), the operators L and J can be understood as particularizations of the operators h and =1, respectively.	■
II.3.1.8. Axions of Informational Embedding
Let, for instanoe, sensory or outward information <r arrive into informational domain of the so-called source information a. In this case, the perception of cr by a is possible only through Informing of <r, namely in the way that a is informed by <r. In this Informing, a is in no way a passive actor, because the state of being informed is in fact Informing within information in the presence of the outwardly appeared cause, i.e. information cr. The acceptance or perception of this Informing is called informational embedding, in general. The nature of informational embedding is to embed the arriving information into the source information. Evidently, embedding in this sense is a dynamic, unforeseeable Informing. Informational embedding explains and illuminates Informing of information from a particular point of understanding.
It is evident that Informing of the arriving information a can only be informational, or that the acceptance or perception of a by a can only be informationally particular. The only exception from this general principle can be observed within the so-called data processing, occurring in traditional, lifeless machines.
r. . .DF12. [Axioas] :
Let a be source information, cr arriving (sensory, outward) information, E informational embedding, and s information embedded into a by E. A series of axioms of informational embedding can be constructed:
(1)	(a |= a) î ((V is embedded into a') A
('a is embedded into cr ' ) )
(2)	(<r h a) * ( ( <r, a L ®) A (E L e))
(3)	(a H <r) ^ ( (E J a, <r) A (e J E) )
(4) ( (cr (= a) LE, e ) $
( (® = ® ) A ( e = E (0-, a)) &,a	a, a
(5)	u , E J (a =) cr) ) i
( (E = E ) A (e = E„Ac, or) )
Ct , <7	OC , <7
(6)	(<r t= oc) S ( (e C a) A (<r £ a) >
(7)	(aha) ^ (e cr, a)
etc. It is evident that the Bo-called axioms of informational embedding can be constructed indefinitely, for instance, by using the principle of particularization, etc.	■
Let us comment the listed axioms. In general, if a informs a, then an informational interaction between cr and a occurs in the form that <r and' a are simultaneously embedded in each other. This fact is easily understood in the case of a living being, where sensory and, for instance, perceptual (or cortical) information influence each other. If a informs a in one (2) or another way (3) , then informational embedding E arises from <r and a and information of embedding £ appears as a consequence of embedding as Informing. If cr informs a in one (4) or another way (5) and this Informing causes the appearance of embedding E and information of embedding e , then in one case (4) the embedding is equivalent to E^. and information of embedding is equivalent to ®ff a> anc> in another case (5)
the embedding is equivalent to ®	and
a , <r
a, a
information of embedding is equivalent to e
If <r informs a (6), then e becomes a part (C) of a, however a is not a part (t) of a. Simultaneously, e is similar to a as well as a (7) .
It is certainly possible to introduce the explioit informational operator of embedding, for instance 1=_, which could be even more comprehensible than its implicit (functional) counterpart, denoted by ^{cr). The meaning of this implicit case is 'or is in the process to be embedded into a by a' .
II.3.1.9. Axioms of Informational Embedding of Counter-Information
In contrary to sensory information, counter-information is a product of information itself. It appears as a Kind of inward sensory information, which has to be perceived by information itself. This se 1f-perception is performed through the process of embedding
r. . ,DP13. (Axioas] :
In thiB axiom we use the following symbols: a is information which informs, however, also informs in itBelf. w is counter-information which comes into existence through self-Informing of information a. Further, C denotes counter-Informing within a and C denotes the process of embedding performed by a. The following axioms of informational embedding of counter-information are only a few of possible ones :
35
(1)	(a h a) * ( (a L w) A (et w>>
(2)	(aHct) ■> (<wJct)A(uH a))
(3)	(a f= a)	( (a ^ w) A (a (=g u>) )
(4)	(a H av S (<u a) A (u a))
(5)	(a L u) ^
((a |= Sa(u)) A (a h «aia(«)))
(6) (u J a)
( (Ka(u) =1 ot ) A <ea(a<"> =1 ot) )
etc. The axioms (3), (4), (5), and (6) bring the so-called cyclic (also circular) nature of information in the foreground, when information is understood as a cyclic process of counter-Informing and embedding of information.	■
The following comments to the last axioms are
possible: If information a informs in itself in
one (1) or another way (2), then, from a or by
a, counter-information u is coming into
existence and this counter-information is
embedded in a in one (1=^) or another way (=^1-
If information a informs in one (3) or another
way (4), then a counter-informs counter-
information u> in one ((=_) or another way (=(_)
»	tc
and embeds counter-information u in one (t^g) or
another way (Hg)• If ot causes the appearance of
counter-information u in one (5) or another way
(6), then information a informs its own
counter-Informing ^(w) in one or another way
and informs its own informational embedding
®a ^w) in one or another way. The last four
axioms constitute the cyclic nature of
arising informational entities.
Vp............°
with the following meaning: a, p, ... , y
differentiate %, T), ... , £ or !;, .......£
are differentiated by a, p, ... , y.	m
rr, . . EX 14 IOperators] :
To clear the meaning of the explicit informational operator of differentiation let us look at the following examples:
a t=s	information a differentiates (or a has
the function of an infomalional differentiation);
t=j) a	information a is differentiated;
o Nj P information a differentiates
information p or information p is differentiated by information a;
«. 0.....T Ç > ......Ç .
informational entities a, p, ... , y differentiate informational entities
11» ••• , Ç or	.......Ç nre
differentiated by a, (5, ... , y etc.
This general case of	informational
differentiation can	certainly be determined also for parallel, cyclical, and
parallel-cyclical cases	and 1-^) . ■
[Axioms]
DF14.
(1)	((a H v (t= a)) * ((a »= ^) V ((=B a))
(2)	(H a) v (a =0) S <HS et) v (a =y )
(3)	(a (= a) * (a 1=^ oc)
II.3.1.10. Axioms of Informational Differentiation
Differentiation of information is an inherent property of information which informs and is informed. Differentiation is not only informational arising, but arising of informational difference in comparison to the present state or processing of an informational entity. Differentiation is a component of informational arising with the intention to arise differently to existing information. The consequence of this fact is that information arises differently. To enter into the discourse concerning informational differentiation, we can introduce two basic and general differential operators which govern the so-called explicit and implicit informational differentiation.
.. ,DF4 5, [Operators] :
The explicit informational operator of differentiation can be determined in the following way:
(4) (a (= P) * (a, p ^ a, p)
etc. The last axiom can be constructed from a more general one, namely from,
(a (= P) * (a, p h a, P)
by the non-uniform substitution of the second operator, i.e., by its particularization.	■
Through informational differentiation of information also several differences can be determined. These differences can be marked by special symbols. For instance,
8 „ „(5, rj.....O s
a,p , . . . , Y
< (ot. P.....y 1= 5. ......K) v
(5. t). • • • i ? =1 ot, p.....y) )
This formula shows the possibilities of conversion between implicit and explicit informational operators, where marking of a difference becomes equivalent to the result of an implicit operation. For instance,
<NS v Hs)
"Df
('differentiates' ) V ('differentiate') V ('is_differentiated(_by)') V ('are_di f ferentiated(_by)')
8a,P.....'.....Ç> » w
VP.....y ".....°
The implicit informational operator of differentiation can be determined as
36
II.3.1.11. Axioms of Informational Integration
Integration of information is an Inherent property of information which informs and is informed. By integration the incoming, arriving, and arising information is informationa11y integrated into source information or into information in question. Informational integration is a consequence of the appearing information which has to be integrated into an existing informational realm, otherwise it will be lost as informational noise. Similarly as in the case of differentiation, it is possible to introduce two basic and general operators of integration which govern the so-called explicit and implicit informational integration.
[ Ope r»tor8]®^^:
One kind of the explicit informational operator of integration can be determined in the following way:
<>=9 V

~Df
('integrates') v ('integrate') v ('is_integratod(_by)') V (*are_integrated(_by)')
The primitive implicit informational operator of integration can be determined as
3 ft	T),
a, P > • . . , Y
. O
with the following meaning: a, 0, ... , y integrate i), ... , ? or t), ... , £ are integrated by a, p, ... , y into a, p, ... , T-Evidently, the operators	Hgt and 3
integrate the given information into the integrating information itself. At this point a clear difference between differentiation and integration comes into the foreground.
Certainly, the complete implicit informational operator of integration can be determined in the following way:
3[X,n.....v]a,0,...,T<?' .......
The meaning of this implicit operator is as follows: informational entities a, (5, ... , y integrate informational entities Vt ••• • C into informational entities X, (jl, ... , v. ■
[Operators] :
Now, we have to define an informational operator of location with the meaning "into", to enable the expression of this particular need, for instance, to be integrated "into" information. This operator has to be of explicit informational type, for difficulties of expressing the process concerning the "into" occur, for instance, by the use of the operator • We have:
and introduces substantial semantics into our further discourse.	■
[Operator]"*15:
In the explicit case the informational operator of integration (=g the need has arisen to express into which informational entity information will be integrated. We have now the following possibility:
a 0 A,
y ß a
The meaning is the following: information a
integrates information 0 in one or another way
into information y. By the operator I we can
even capture the most subtle phenomenon of
coming of information into existence. Thus, we
can decompose the operator L to some degree by
splitting its meaning into "coming of' <t= )
~~	come
and 'into'. We can introduce the following implication:
<L <x)
(	a) I a)
come
At this point the question what is existence can arise. "Existence" has the meaning of information of existence or of existing information. Similarly, "coming" has the meaning of information which comes or of coming information. This kind of information is, for instance, coun ter-information or sensory information. Coming of information into existence is, for instance, embedding of the arisen counter-information into the existing information. Here, coming into existfncp concerns informational differentiation as well as informational integration. In other words, arising of information is nothing else but counter-Informing and embedding or differentiation and integration of information. These two types of Informing constitute the so-called informational cycle, which is the cycle of coming into existence: from the existing information arises the counter-information and is embedded again into the existing information, enlarging (or decreasing) its informational realm. This informational cycling is the fundamental process of any informational arising. Therefore, we can say that information informs (differentiates) and is informed (integrates) cyclically or, in a more general sense, circularly.
[Operators]KX16:
Let us clear the meaning of the explicit informational operator of integration in composition with the informational operator 'into'. We can list the following examples!:
« K
«=3
~Df
('into')
or (ct (=3> 1 a
information a integrates or information a integrates into itself;
r ((=g a) X o
information a is integrated or information a is integrated into itself;
Further, we can introduce the left to the right and the opposite version of this operator by
X„ and X
respectively. The proposed operator is general
«KP
(a Kj P) XT
information a integrates information 0 or information a integrates information p into information 7; the last formula can also be read as i nf orna t i o'n P is
37
integrated by information a into information t;
ct, p, ... , T (=3 Ç, T), ... , Ç or (a, p, ... , T l=3 ......Ç) A
A , (i, ... , v
Informational entities a, P, ... . "T integrate informational entities Ç, t), ... , Ç or informational entities a, p, ... , f integrate informational entities Ç, t), ... , Ç into informational entities X, (jl, ... , v; the last formula can also be read as informational entities i), ... , Ç are integrated by informational entities a, p, ... , f into informational entities A, u, ••■ > v, etc.
These cases of informational integration, using the explicit operator |=g, can certainly be
determined also for parallel, cyclical, and parallel-cyclical cases (l=g> Hg, and H"a).	~
.DF15.
[Axioms!
(1)	((a (=) V	<H	a)) 4> <(oc Hg) v (h=3 a))
(2)	(H a) V	(a	=0) * < < =4^5 a) V (a
(3)	(a |= a)	*	<<a K,	a) * «) )
(4)	(a KPI	* <<*> P	Kj «. P> i <*' P > >
etc. The last axiom could be constructed from a more general one, namely, from
(a |= p) * Hot, p h cc, P) N a, P))
by the non-uniform substitution of operators (e.g. particularization of the second operator on the left side of implication).	■
[AIIobbI®:
The axiom (4) in the laBt definition can be decomposed in details in the following way:
( ( (a		a)	JL	a)	V
((a		P)	1	<*)	V
<(P	•=3	a)	I	a)	V
UP		P>	1	a)	V
<<ct	hs	a)	1	P>	V
((«		P)	Jt	P>	V
UP	>=3	cc)	X	P)	V
UP		P)	1	P>>	
This ' is the well-known principle of the iwff decomposition. The so-oal1ed parallel decomposition of the last case would be as follows:
II.3.12. Axioms of Informational Particularization and Universalisation
In informational logic iwffs can be particularized and universalized. This principle permitB various substitutions of explicit operators, enabling specialization (particularization) and generalization (universalization) of informational formulae. Processes of informational particularization and universalization are the basic, i.e. axiomatic properties of an iwff. These processes could be included as well into the domain of the so-called transformation rules, for through their application, formulae are transformed from original semantic domains into other special or general ones.
[Axioms]
DF16.
<1) ('a is iwff')
(2) ( '(= is sub-iwff ' ) ' con
<"5S(a) is iwff')
() is sub-iwff') con
(3)	(at (= P) * («	P)
(4)	(P =| a) * <p <PH> a)
(5)	V<a (= P)	' <*<a> H(H ®(P) )
(6)	$(P =|a| => <*S(p> <CH) 5(a))
etc. In fact, particularization is an implicit informational operation. Further, the symbol $ can be used for particularization ) as well
t
as for universalization ($ ) of formulae. Particularization iB always a non-uniform operation in regard to substitution of operators. By particularization and universalization new semantics of operators and formulae is generated. Particularization and universalization belong to the most essential principles of informational arising.	■
II.3.13. Axioms of Informational Formula
We have already determined the so-called formation rules of iwffs. However, this rules do not ensure the constructibility of a formula which haB to interpret a natural or an artificial information. We would like to know, at least hypothetically, if we do not need to take care about the nature of information which has to be formalized or put into the form of an iwff. Thus, the following questions may sound quite naturally: What kind of information can be put into the form of an informational formula? How can information be put into an adequate form of a formula? Is this form in case of information a unique or a multiplex
(a (= P) * ((a &=3 a) X
(a |=3 p)	1	ai
(p |=3 a) X a, (p K, P)	X	a,
(a |=g a) 1 p, (a p>	1	P,
(p 1=3 a) X p, <P 8=3 P>	1	P)
This example is in fact the axiom of the parallel decomposition of the case a M-	■
[Axioms) :
Let a denote an arbitrarily complex information. Let informational formula be denoted as an informational1y well-formed formula (iwff). Then the following basic axiom is adopted, concerning the possibility of forming an informational formula from given information :
38
(Va).(('a is information') i> < 3{'iwff).
('a can be put into the for« of aa iwff')))
ThiB axiom says: for each a, which is information, irrespective of its complexity and informational nature, there exists at least one iwff such that (•) information can be put into the form of this iwff.
(Va).(('a is information') £ (3 ( ' iwff ) .
<'iwff is an adequate interpretation of a')))
This axiom says: for each a, which is information, irrespective of its complexity and informational nature, there exiBts at least one iwff such that this iwff is an adequate interpretation of information in question. We can understand that formal interpretation of a given information is never unique and that it depends on informational circumstances. In general, there exist (indefinitely) many interpretations of a given information.
(Va).(('a is information') ^ (3('iwff').
('iwff interprets a by an informational system of one or several sequences of informational operands and operators')))
This axiom assures the constructibility of the iwff which interprets adequately the given information a.
These three axioms can be certainly expreBBed in a much more symbolically compact form, for instance:
(Va).(((a )=) V <*= a)) (3,.).((a V) \
<3 »i>»2.....^n* • <^1 ■''2'" ' ^ a>)>
((3?).((a <t) V (9 Had0 <*)) V
where 9 symbolizes iwff, =4a(je is informational
operator of adequate interpretation, and each of the three conductive parts on the right Bide of implication concerns one of the three ax i oms.	g
II.3.14. Axioms of Informational Structure and Informational Organization
What are informational structure and informational organization and how do they reflect in informational axiomatization ? To answer this question we have to consider the principle of informational structure and informational organization ([4] or [10]) as follows :
"Informational structure is a constitution of information, that is, a constitution of informational forms and informational processes that are composed as information. These forma and processes are informational components. The informational relations among informational components which determine a composite: information constitute informational
organization. In terms of informational epistemologjr, informational structure is closer to the form, where*» informational organization ia closer to the procesm. Within information, informational forms and informational processes are informationally interwoven components. Informational components integrate information. Informational structure and informational organization are information by themselves."
What are forma and proceBBes constituting a particular informational case? Speaking in the language of informational formulae, these forms and processes are informational operands and operators, where forms are, for instance, self-standing operands and processes are formally grouped (parenthesized) operands and operators. In this respect, informational structure appears aB a more or less pure syntactic structure.
We suppose that given information always has a structure. This structure, which is observed as information concerning the structure of information in question, can always be interpreted through an iwff in a simple informational case or through an informational system of iwffs in a complex informational case. The structure of information is interpreted in iwffs by particular informational forms and processes, consisting of informational operands and operators. It is possible to list several axioms concerning the structure of information, for instance:
[Axioms J
(1)	(Va).(('a is information') *
((3cr).('<r interprets the syntactic nature of information a')))
In this axiom, <r denotes information concerning the structure of information a.
(2)	(Va).(('a is structured information') *
((3ip).('a is interpreted by an
adequately syntactically structured <(>')))
In this and in the next axioms, <f> marks the so-called iwff.
(3)	(Va).(('a is information') »
((3<p).('<(> as information
syntactically constitutes information a')))
(4)	(Va).(('a is information') $
((3<p).('<p interprets the syntactic Btruoture of information a')))
etc. These axioms constitute the so-called structural hypothesis of information. This hypothesis, in fact, is the informational principle of structural constructibility of information and its adequate iwff. Here iwff iB understood to be an informational system or any form of informationally connected iwffs.	■
The structure of information a iB information which concerns the coaponential syntax of a.
39
This syntax is interpreted into the structure of iwff of a. Structural interpretation of a onto its iwff does not represent the sufficient condition for a completely adequate interpretation of a by its iwff. The second component, called informational organization of a, has to be considered when the iwff adequate to a is constructed.
While the structure of information predominantly concerns the so-called syntax or form of componential constitution of information, organization of information predominantly concerns the so-called semantics or componential processes, relations, and informational interweaving of informational processes. Among possible interweaving of informational forms and processes within information, the most important seems to be informational parallelism, which is the synonym for interweaving nature of informational forms and processes. It becomes evident that information, by its nature, is nothing else but extremely interwoven structure (topology, granularity) and organization (selectivity, relationship) of information.
[Axioms]
DF19.
(1)	(Va).(('a is information') $
((3u).('u interprets the semantic
nature of information a')))
In this axiom, u denotes information concerning the organization of information a.
(2)	(Va).(('a is organized information') $
((3<p).(('a is interpreted by an adequately semantically organized cp' ) v ( 'cp semantically interprets a as informational organization'))))
In this and in the next axioms, 9 marks the so-called iwff.
(3)	(Va).(('a is information') ï
((3<p).('<p as information
Bemantically constitutes information a')))
Instead of the consequence in the last implication it could be
a Nj t 6(a) or, conventionally, a = 6(a)
6(a) has the meaning of 'syntactical, i.e. structural nature of a', whereas the operator 1=^ t has the meaning of Informing by interpretation.
(3)	('a is structured information') >
( (a (= ct) A (ct C a) )
If information a is structured, then it informs (gives, transmits) information ct of its structure (c C a).
(4)	('a is interpreted by an adequately
syntactically structured 9') $
( (<r C a) A (ct C 9) A (9 £ n ot)>
The iwff 9 informs structurally
(syntactically) similar (analogous) to
information a. The iwff 9 informs
structurally similar (by means of the
operator	) to a.
r	syn
(5)	('9 as information syntactically
constitutes information a') 5 (((a C a) $ (a C 9)) A
(3(9 t=).<9 t=„yn «>))
The iwff 9 in fact constitutes also the structure of information a. The operator of this syntactic constitution is k-
syn
There exists such Informing of 9 that 9 syntactically informs a.
(6)	('9 interprets the syntactic structure
of information a')
((ct C a) * (9 Hsyn a))
(7)	('w interprets the semantic nature of
information a') $ (u h 51(a))
Instead of the consequence in the last implication it could be
m t. . 8(a) or, conventionally, u = ï(a) int
a(a) has the meaning of "semantic, i.e. organizational nature of a".
(4) (Va).(('a is information') *
((39).('9 interprets the semantic
organization of information ce')))
etc.
[Axioms]
BX3.
DF18	DF19
The axioms [Axioms]	and [Axioms]	can be
interpreted in a more symbolically compact and
instructive manner. Let us construct the
following implications:
(1)	('a is information') ï ( (a (=) V (>= a) )
(2)	('<r interprets the syntactic nature of
information a') $ (<r 6(a))
(8)	('a is organized information') $
((a)=«) A (w C a) )
If information a is organized, then it informs (gives, transmits) information u of its organization (u C a).
(9)	(('a is interpreted by an adequately
semantically organized 9') v ('9 semantically interprets a as informational organization' ) ) =>
((u C a) A (u C 9) A <9 «Bem «))
The iwff 9 informs organizationally (Bemantically) similar (analogous) to information a. The iwff 9 informs organizationally similar (by means of the
40
operator .". ) to a. ^	Bern
(10) ('9 bb information semantically
constitutes information a') £ <((w C a) * <w C 9) ) A (3(9 N).(<P f=8eB a)))
The iwff 9 in fact constitutes also the
organization of information a. The
operator of this syntactic constitution is
k . There exists such Informing of 9 rBem
that 9 semantically informs a.
The consequence of this system is a N 9. The construction of iwff 9 from a is a parallel informational system which assures the so-called formalization of information a onto iwff 9• ' Thus, the last system can be particularized in the form
a Mi a }= u>, a, u f= 9
The consequence of this system i s a f= <p . Further particularization is possible:
(11) ('9 interprets the semantic organization
of information a') ^
((w C a) i> <9 (= _ a) ) sent
DP 18
Considering implications (l)-(ll), [Axioms] D F1 9
and [Axioms]	can be rewritten in the
following manner:
(1)	(Va).(((a H v a)) ^
( ( 3<r ) . ( cr (= 6(a))))
(2)	(Va).(((a t= o-) A (<r C a)) i>
(09).((a C a) A (<r C 9) A (9 A a))))
(3)	(Va).(((a t=) V (N a)) i> ((39). (<(ct C a) ^ (<r C 9)) A
(3(9 (=>•<* N=Byn «))>)
(4)	(Va).(((a |=) v (|= «)) *
((39).((cr C a) ^ (9 t=syn a))))
DF19
etc. Further, for [Axiomsl	there is:
a k a, ot	to,
syn	sei '
CT , W |=
form
The consequence of this system is a l=forB <P ■
II.3. 15. Axioms of Informational Parallelism
Information is a parallel informational phenomenon in itself as well as in its interaction with other or outside information. It means that its forms and its processes appear, inform, change, vanish, etc. in a parallel manner. In this phenomenology, parallelism can be understood not only topo1ogical1 y and temporally, but also symbolically, abstractly, expressively. The basic question might be how information is performing parallel in itself. Why is informational interaction in fact always n parallel Informing? Thus, these conclusions (or beliefs) can be axiomstical1 y framed in the following axioms:
[Axioms]
DF21
(1) (Va).(((a (=) V (»= a))
(<3<j).(w «(a))))
(1) ('a is information') £
(I (a |=) V (}= a)) V ((4 a) v (a -1)))
(2)	( Va ) . ( ( (a |= w) A (u C a) ) *
<(39).((u C a) A (w C 9) A <9 .'.g^ a)))))
(3)	(Va).(((a H) v (k a)) *
( (39) .< ( (u C a) t> (u C 9) ) A
(3(9 H.(<P h=8en <*)))))
(4)	(Va).(((a H v (|= a)) :>
((39).(<(u C a) $ (u C 9)) A
(3(9 h).(l>	a)))))
If a is information, then it informs and is informed in parallel in one or another way. This fact can be expressed also in the form of parallel informational system, i.e. ,
('a is information') :>
(a |= a, 4 a, a HI)
(2) (Va).(('a is information') >
(3(CC,®).(S |=, |= S , « ft, fr « ) ) )
a a a	a a	a
etc.	■
DF18	DF19
The axioms (Axioms]	and [Axioms]	assure
the existence of an adequate (informationally
complete) interpretation of any information a
onto its iwff 9 in the sense of informational
structure <r and informational organization u.
This leads to the fundamentally important
axioms of construetibi1ity of iwffs for
arbitrarily occurring informational cases.
.. . -DF20. [Axioms] :
The axioms which follow govern . the interpretation of information a onto the structure <r and organization to of an iwff (or of a system of iwffs) 9, which models a in the informationally complete way. The process of constructing iwff from given information can be expressed in the following manner:
If
is information, then there exist
a (= <r,
a w, <r, u t=- 9
counter-Informing caused by a, d , and
a
informational embedding of a, E , which
a
inform in parallel. This Informing of and ®a is an immanent property of parallelism of information. Ab we have already recognized, counter-Informing and embedding of counter-information constitute the so-called basic informational cycle (informational cyclicity). It also follows from the last axiom that counter-Informing and informational embedding within information a perform as information.
(3) (Va).(('a is information') $
<3(Sa, ®a).<<* 1= Va>>
a, S (a) 1= « (K (a)))))
Œ	a u
41
where £ (a) is in fact counter-information a
w produced by counter-Informing S and ®a(Sa(a)) is the embedding of the produced counter-information to into a.
(4) The most complex informational system of inward informational parallelism can be axiomatized by the following iwff:
(Va).(Ca is information') ^
((a, 3 , I , to, « a, 3 , <E , u, <£ ) V a a	a	a a	at
(a, 3 , S , u, « HI a, 3 , S , u, ® ) ) ) a a	a	a a	a
^ The parallel decomposition of the first disjunctive iwff part on the right aide of Implication is
a	1= a.	a	1=	V	a	»=	s , a '	a	IN	u,	a	h	e a
3 a	b <*,	3 a	h	V	3 a	1=	« , a *	S a	»=	w,	3 a	¥	e a
S a	IN a,	C a	IN	9 , ot	S a	IN	<£ , a	« a	IF	u,	S a	h	e a
u	a,	u	8=	V	u	»=	« , a	u		w,	u	IN	« a
® a	»= «,	« a	h	V	® a	IN	a'	e a	1=	u,	« a	IN	® a
(5)	<ct N=	P)	\	3	(a,	P	1= a,	P)					
ThiB axiom determines the so-called inward informational cycle. The formula can be universalized in the following manner:
(3)-	(Va).(('ot is information') £
((((a h Sa) h (a, Ca h w)> b
(a, sa, u I- ®a)) h (a, «a, u, ®a h a ) ) )
In this formula it is possible to observe distinct cycles, i.e., also cycles within cycles, where for the right Bide of implication there is
(((cyclej I- cycleg) (- cycle^) |- cycle^)
In this expression there are three more cycles, namely cycle,, between cyclej and cycleg, cycleg between cycleg and cycleg, and cycle7 between cycleg and cycle^. All these eye les can be understood as cyclically parallel, thus:
(4)	(Va), (('a is information') $
(a I- «a, (a, > w). («•	w ^ ®a>>
( a , S , w, ® I- a ) ) a	a
etc.
ThiB cyclic parallelism can be captured in the moat complex form by the iwff
II.3.16. Axioas of Informational Cyclicity
Information is a cyclic informational phenomenon in itself as well aa in its interaction with other or outward information. It means that informational forms and informational processes appear, inform, change, vanish, etc. in a cyclic manner. Cyclicity of information can be viewed to be purely serial, parallel, or serial-parallel phenomenon. The last case seems to be the moat obvious one. Within this phenomenology, cyclicity can be understood not only topologically and temporally, but also symbolically, abstractly, expressively. The basic queatlon is how information.performs cyclically in itself. Why informational interaction is in fact always a cyclic Informing? Let us frame these observations axiomatically In the following manner :
r. . .DF22. [Axioms] :
(1)	('a is information') >
(((ah) V (J- a)) v ((a R v <»- a)) V (Ha ) V (a H)) V ((HI a) V (a -J) ) )
If a is information, then it informs and is informed cyclically and parallel-cyclically in such or another way. This fact can be expressed also in the form of parallel-cyclical informational syBtem, i.e., in a particular form:
('a is information') ^
(-1 a, a -), HI a, a HI)
(2)	(Va).(Ca is information') ^
( (a h SŒ) f\ (a, «a (= u) A (a, Sa, u h ®a) A (a, Sa, <j, ®a t= a)))
(5)	(Va).(('a is information') *
(a, S , w, « I- a, «£ , w, « ))
a	a	a	or
(6)	To explain the nature of informational
cycle, the following auxiliary axioms can be adopted:
(£ f a) 1 w and (« |- w) 1 a
a	a
with the meaning to = K^fa) and ®a<u) c a> respectively.
Obviously, axiomatization of informational cyclicity can be continued indefinitely.	■
II. 3.17. OpennesB of Informational Axiomatization
But taking the methodologies as an end in themselves is ultimately limiting in the same sense as the analytic tendency to take the arguments as an end in themselves.
Terry Winograd 112] 255
The axioms determined show the possibilities of their indefinite axiomatic continuation. Beside the already existing axiomatic cases new axiomatic interpretations are possible which concern an axiomatic type. In a similar way it iB possible to add new axiomatic types to the existing ones. The consequence of these possibilities iB that an axiomatic syBten remains open for new axiomatic determinations. Finally, it is possible to conclude that informational axiomatization irrespective of the informational system involved remains open in the described sense. To clear this informational phenomenon to some extent, we can put several principled questions concerning the
42
structure, organization, parallelism, etc. of i n f ormat. i on .
The axiomatic basis of informational logic remains open. Principles of informational particu1 arization and universa 1ization contribute to an additional and constructively senseful component of keeping the axiomatic basis open. In fact, informational logic in its axiomatic nature performs as regular information. Thus, the exposed axiomatization in this essay ia informational.
II.3.18. Informational Axioms and Metaphysical Beliefs
We want to expand our ability as observers, within a context in which we are not detached but are engaged in the practices we ourselves observe.
Terry Winograd [12] 255
It cannot be disputable that the listed informational axioms arise from a particular metaphysical disposition from which they are thrown into a broader professional, scientific, and certainly also philosophical discourse. Whichever theory comes into existence, it begins its march a8 a scientific or philosophical literature and in fact represents nothing more than an authorial telling of a story. This storytelling, which concerns informational axioms and processes of axiomatization of diverse informational principles, grounds in epoch-making beliefs, i.e. in the metaphysical background constituting the philosophy of the so-called information era. Again, metaphysics has to be understood as a totality of information spontaneously arising in a living being and within its population.
The awareness that axiomatization of informational principles grounds in metaphysical beliefs lets the processes of axiomatization be generative, indefinitely predictable, and open for further development. Such kind of axiomatization certainly does not fit properly into the hardly predestined realms of traditional and emphasized rationalistic science. Does the time come when new, non-traditional, and also non-rationalistic approach in exact sciences is becoming an evident advantage in the research of unrevealed possibilities?
II.3.19. Some Axiomatic Consequences of Informational Arising
but also in implicit informational operators and informational operands. By themselves, axioms are arising structures of informational formulae. In this respect the axiomatic consequences of informational arising can f i r><! their continuation in any construction of iwff.
References of IL I, II, and III
[1]	H. L. Dreyfus and S. E. Dreyfus: Mind over Machine. The Free Press, Macmillan, New York 1986.
[2]	A. P. Zeleznikar : On the Vay to Information. Informatics 11 (1987), No. 1, 411.
[3]	A. P. Zeleznikar: Information Determinations I. Informatics 11 (1987), No. 2, 3-17.
[4]	A. P. 2eleznikar: Principles of Information. Informática 11 (1987), No. 3, 9-17. [Published also in Cybernetica 31 (1988), 99122. ]
[5]	A. P. Zeleznikar: Information Determinations II. Informática 11 (1987), No. 4, 8-25.
[6]	A. P. Zeleznikar: Problema of Rational Understanding of Information. Informática 12 (1988), No. 2, 31-46.
[7]	1,. A. Steen (Editor): Mathematics Today. Twelve Informal RBsays. Springer-Verlag. New York (Third Printing, 1984).
[8]	E. de Bono: The Mechanisms of Mind. Penguin Books, Harmondsworth, Middlesex, England (Reprinted 1977).
[9]	G.W.F. Hegel: The Science of Logic. In Hegel'b Logic, being Part One of the Philosophical Sciences (1830), translated by W. Wallace. Oxford, At the Clarendon Press, reprinted 1985.
[10]	A.P. Zeleznikar: Prlnci pies of Information. Cybernetica 31 (1988) 99-122.
[11]	S. Schiffer: Symposium on Remnants of Meaning: 1. Overview of the Book. Mind and Language 3 (1988), No. 1, 1-8 (Basil Blackwell) .
[12]	T. Winograd: On Understanding Computers and Cognition: A New Foundation for Design (A response to the reviews). Artificial Intelligence 31 (1987) 250-261.
At the end of section II. 3, in which we have discussed informational axioms, it seems necessary to stress again the arising (or at least variable) nature of informational operands, operators, and formulae. Such as they are, all of the listed axioms in this essay concern the arising principle of occurring informational entities. Thus, this informational nature is found not only in the semantics of explicit informational operators