ELEMENT S OF METAMATHEMATICA L AN D INFORMATIONA L CALCULUS 
Anton P. Zeleznikar An Active Member of the New York Academy of Sciences Volaričeva ulica 8, SI 61111 Ljubljana, Slovenia 
Keywords: axioms of parallelism, serialism, circularity and spontaneity; catalogue of informatio­nal rules; decomposition axioms, decomposition theorem, deduction theorems, implicative axioms, inference rules; informational calculus, predicate calculus, propositional calculus; substitution rule; various axioms and theorems (implicative, conjunctive, disjunctive, equivalent, negatory, etc.) 
Edited by: L. Birnbaum Received: November 4, 1994 Revised: January 11, 1995 Accepted: February 3, 1995 
This article deais with problems pertaining to the elements ofa.xioma.tics in traditional (mathematical, symbolic, philosophical) logic and to the problems ofnewly emerging axi­omatics in informational logic. Informational axioms can be derived from propositional and predicate axioms and then, particularized and universalized within the informatio­nal domain. Traditional axiomatic formulas of the propositional and predicate calculus can become a rebounding cause for the construction of essentially different axioms in general informational theory. It is shown how propositional and predicate axioms and rules can be informationally extended for the needs of the general informational theory. 
1	 Introductio n 
Which are the basic and inferential informational ajdoms1 which govern the derivation of formu­las (theorems, consequences), that is, their pro­ving procedures (proofs) in informational theo­ries? This question is signiflcant for the consci­ousness of tha t what scientific theories do out­side of theories themselves, in their—from theo­ries themselves separated—metatheories. On the other hand, informational theories are always uni­ons of object theories and metatheories, where the last have the role of being the producers of theo­ries in the sense of informational arising, that is, spontaneity and circularity. 


2 Fundamenta l Figures of Syllogistic Inference 
Syllogistic inference can be interpreted in different ways, for instance, in the scholastic (philosophi­
•"This paper is a private author's work and no part of it may be used, reproduced or translated in any manner whatsoever vvithout vvritten permission except in the čase of brief quotations embodied in critičal articles. 
cal), informational and traditional-mathematical manner. 
2.1	 An Informational Interpretation of Syllogism 
Syllogism (av\\ojrj, in Greek, means ga­thering, collecting, assembly, concourse and av\\o'yi(ofica means to reckon, consider, think, reilect; to infer, conclude) (in German, das Zu­sammenrechnen, der Schlufj, die Folgerung) is a valid (e.g., true) inference in a syllogistic form. 
Syllogistics (in Greek, avXXojiaTiKfj Tixvf) the art of inferring, concluding) was founded by Aristotle and developed in scholasticism as tea­chings of (correct) inference in syllogistic form. This technique became the keystone of the tradi­tional logic ([4], pp. 407-409). 
The inferring in a svllogistic form proceeds from two premises to one conclusion. Thus, the valid inference, the so-called syllogism, with true pre­mises, delivers a true conclusion. In parallel to the traditional logic, as premises of syllogisms, only the following forms of 'equivalent' informational formulas are allowed: 

1.	 "Ali x are F(x)? Formula x \=VxF(x) 
is a universal afnrmative judgement and re­ads, informationally, x informs for ali x the property (entity) F(x). 
2.	 "No x is F(x)." Formula x \L\/xF(x) 
is a universal negatory judgement and reads, informationally, x does not inform for ali x the property (entity) F(x). 
3.	 "Some x are F(x)" Formula x \=3x F(x) 
is a partial afnrmative judgement and reads, informationally, x informs for some x the pro­perty F(x). 
4.	 "Some x are not F(x)" Formula x \Č3X F(x) 
is a partial negatory judgement and reads, informationally, x does not inform for some x the property (entity) F(x). 
The connections among these four formulas can be presented by the logical quadrate in Fig. 1. 
contrary X \=Vx F(x) X ^v * F(x) 
sub- sub- 
al- al­ 
ter­ ter­ 
nate  nate  
CD  

X \=3x F(x) 	X \Č3x F(x) 
contrary 

Figure 1: The logical quadrate, dramn according to the informational formulas of svllogistics. CD means contradictory. 
Four fundamental figures of the syllogistic in­ference can be distinguished, where |=„, |=f, and |=c are determined by 
A.P. Zeleznikar 
l=ci,N,|=c L {\=Vx,\Lvx,\=3x,\/:3x} 

The position of the so-called middle entity [i is decisive and it must appear in both premises in the following manner: 
(FF1) 	(FF2) 
a\=cir 	<7(=C7T 

(FF3) 	(FF4) 
a |=C TT 	a |=c 7T 

If operators |=a, |=f, and |=c are replaced through concrete (particularized) operators |=va;, ^v ^ 1=32; and \^3X, for every čase an inference modus is obtained. In ali, there are 43 = 64 modi for each fundamental inference figure, that is, 256 figures. Only 24 of them are valid, that is, syllogisms (6 for each fundamental form, and for some additio­nal suppositions have to be introduced). A modus is uniquely determined if the fundamental figure and the operators |=0, f=b and |=c are known. 
2.2	 A Mathematical Interpretation of Syllogism 
In modern logic, syllogism is treated in the fra­mework of the first order predicate logic. The so-called universal and existential qauntifier, V and 3, can be represented with conjunctive (A) and disjunctive (V) logical connectives within a cer­tain domain (set) D of elements. E.g., the schola­stic modi "Barbara" and "Felapton", where the emphasized a stands "for ali", e for "no", and o for "some . . .are not", become 
•+ P ( x ) \
xeD 
A s(x) -* M(x) \xeD J 
and A s(x) --> P(x) 
( A M(z)--P{*) \ xLD 
-, S(x)
A M(x) • \xeD J V S{x)i\ P(x) x€D 
respectively. The premise f\ M(x) in Felapton 
must be completed. Predicate P(x) corresponds to the middle entity /d in fundamental formulas (FF1-4) and, adequately, predicates P(x) and S(x) correspond to entities 7r and a in (FF1-4). 
The reader can find the logic quadrate for for­mulas, together with negatory cases, using univer­sal and existential quantifiers and predicate F{x) in Fig. 2, 
contrary 
VxF(x) 	3xF(x) 
sub- sub- 
al- al­ 
ter­ ter­ 
nate  nate  
CD  

3xF(x) 	VxF(x)
contrary 
Figure 2: The logical quadrate, drawn according to the predicate formulas of syllogistics using quan­tifiersV and 3. CD means contradictorv. 
In Fig. 2 the contrary, contradictory and subaL ternate cases are shown in a clear, that is, nega­tory manner. There is, certainly, 
sixF{x) = 3xF(x) and 3xF{x) = MxF{x) 
and this equivalences (see Subsection 3.7) can be used as interpretations in Fig. 3. 


 A Short Overview of Some Mathematica l Axioms 
In der Aussage: „Der Hammer ist zu schwer" ist das fiir die Sicht Entdeckte kein „Sinn", sondern ein Seiendes in der Weise seiner Zuhandenheit. ...Aussage besagt soviel wie Prddikation. Von .einem „Subjekt" wird ein „Pradikat" „ausgesagt", jenes wird durch dieses be­stimmt. Das Ausgesagte in dieser Be­deutung von Aussage ist nicht etwa das Pradikat, sondern „der Hammer selbst". 
Martin Heidegger [2] 154 Informatica 19 (1995) 345-370 347 
contrary 
\fxF(x) 
yXF{x) 
sub-	sub­
al-	al­
ter-	ter­
nate 	nate 
CD 
3xF(x) 	3xF(x)
contrary 

Figure 3: The logical guadrate, drawn according to formulas in the previous picture with the resolved negations of the guantified formulas. CD means contradictorv. 
3.1	 Introduction 
What is the nature of mathematical axioms and which kind of axioms are significant for our di­scussion? As we shall see, propositional axioms can serve as an outlook to the axioms used in the predicate calculus and also in the informational approach. The beginning of the general informa­tional theory (GIT) has to be founded in logical axioms by which the informational phenomena­lism (externalism, internalism, metaphysicalism) becomes a consequence of the very first assump­tion, that is, of the informational entity. 

3.2	 Axiom s of th e Propositiona l Calculus 
Aussage bedeiitet primar Aufzeigung. ... Aussage bedeutet Mitteilung, Hera­ussage. Als diese hat sie direkten Bezug zur Aussage in der ersten und zweiten Beduetung. Sie ist Mitsehenlassen des in der Weise des Bestimmens Aufgezeig­ten. 
Martin Heidegger [2] 154-155 

Axioms of propositional calculus are fundamen­tal for (the entire) mathematics (metamathema­tics) so that they can be reasonably extended, for example, to different predicate calculuses, ari­thmetic and mathematical logic in general. There are sever al "systems of axioms" which differ, thro­ugh tirne, from one author to another. In our approach, the axiomatic systems of Hilbert [3] and Novikov [5] were chosen. 
In propositional calculus, axioms can be grou­ped in a traditional way (e.g., following Hilbert and Bernays [3] and Novikov [5]). According to [3], p.66, the initial axiomatic formulas can be grouped and, roughlv, written in the form of axi­omatic rules: 
I. Axioms of implication 
1) A-(B->A) , 
2) (A - (A -B)) -> (A -B), 
3) (A-JB)-(OB->C)-(A-C D 


II.	 Axioms of conjunction 
i) AAB^A, 
2) AAB^B, 
3) (A-+B)^> 

((A^C)^(A^BA C)) 

III.	 Axioms of disjunction 
1) A-*4V5 , 
2) 5-4AV5 , 
3) (A^C)^ 

((B -> C) -• (A V B -• C)) 

IV.	
 Axioms of equivalence 1) (A = 5 ) - (A - 5) , 2) (A = 5)-(5-A) , 3) (A-5)-((B-A)-( A = 5)) 

V.	
 Axioms of negation 
1) (A -+_0) -+ (B -»• A) , 
2) A^Z , 
3) A-* A 



The presented system of propositional axioms is not the only possible. For instance, Novikov [5], 
p. 75, chooses the group I of implication axioms in the form 
1)	 A ^( B ^ A), 
2 )	 (A - (A-> C»-> «A - 2?) - (A - C]) 
Various axioms can be useful for the interpreta­tion of basic informational cases of phenomena­lism, as we shall see in one of following subsecti­ons. 
3.3	 Inference Rules of the Propositional Calculus 
Inference rules of propositional calculus are con­structed on the basis of propositional axioms in the preceding subsection. It is hard to say which A. P. Zeleznikar 
"rules" are the primarv, the axiomatic or the in­ferential. However, it is clear that inferential ru­les must strictly consider the primitive axioms of propositional calculus which are, for example, im­plicative, conjuctive, disjunctive, equivalent and negatory. 
Which is the general philosophy of making (constructing) arule, precisely, the inference rule? Irrespective of the theory, for which rules are con­structed, these are always implicative, although they express something more than a pure impli­cation, because they concern the so-called deriva­tion procedure. The role of a rule in the deriva­tion procedure is the folknving: taking a rule in which several premises are logically connected in one or another way, some of them can be deta­ched in the form of the so-called conclusion and, thus, between the respective premises of the rule and its conclusion a special operator, marked by h is used in propositional and predicate calculus while in the informational calculus we use opera­tor -» for marking derivation and an operator (­for marking the circular form of informing. 
Let us settle the general form of an inferential rule for deriving propositional formulas from axi­oms or from already deri ved formulas. 
The iirst rule is substitution. Let us mark propositional formulas which depend on various propositional variables by the capital Fraktur letters, for example, as 21 or, in more detail, 2l(Ai, A2, • • •, An). Let S mark the operator of substitution (in informational terms, the function of substitution). Let us introduce 

s;i 1^^Ba(Ai,A2,...>AB) = 
^(•••Sj2 2(s51 12l(Al!A2,...,An))...) 

where Q5i, ©2, • • •, 2J„ are propositional (identi­cally true) formulas. Thus, the substitution rule has the form 
gi(A1,A2,---,Ara) 
s;i 1 l^.;xfBa(Ai,A2,...,An) 

The second production rule is applied to a formula which is structured as a parenthesized sequence of implications, that is, 
2ii-*(si2->(---(a„-i-»»«)•••) ) 

and is expressed in the following manner: if for­mulas 2li, 2l2, • • •, 2ln_i and 
ai->(a2 -.(---(2in _i-*sin )---) ) 

are true then formula 2l„ is true in the propo­sitional calculus. The complex rule of inference (modus ponens) is 
2li,5l2,---,Sln-i , 

In general, we have the following scheme of infe­rence, where ^3; are true premises and <Lj are true conclusions too: 
In čase m = n, the senseful inference scheme be­comes 
fflQQl, <Ll), ff20Q2, g 2) , • • •,ffnQQn, Cn) 
€l, «2, •••,«» 

where conclusions 
Li,Č2, ••-,<L„ 

are detached from premises 
Vi(Qi , <ti), ^2(Q 2, (L2), • • •, #»CQn, C„) 

and, according to the convention of derivation for­mulas, there is 
O i h d , L 2 h Č2, • • •, & n h €„ 

or, even more generally, 
LJi, L22> • • -, Qn H Ci, € 2, ••-,€ « 

3.4	 A Theore m of Deductio n withi n th e Propositiona l Calculu s 
Formula 93 is derivable from formulas 2ti, 2l2> • • •» 2ln, that is, 
Sli, 2l2, • • •, »n I" » 

if it is possible to derive formula 35 merely by means of inference rules, using the initial formulas 2lii 2I2, • • •. 2l„ and any true formulas within the propositional calculus. 
Informatica 19 (1995) 345-370 349 

Deduction Theorem. If formula !B is derivable from formulas 2li, 2I2, • • •> 2l„, i/zen 
2li-(5l 2-(--•(«„-»)•••)) 

is a true formula. • 
By induction it is possible to prove the following: if 
2l1,2l2,"-,2lIl_1,2ln h » 

then 
Sli, 2t2, • • •, 2ln_! h 2l„ - <B 

where symbol h is used for marking a derived for­mula within the calculus. 
3.5	 Axioms of the Predicate Calculus 
If TI is a set of objects and a,b,c,d are elements of 971, then P(a), Q(b), R(c, d), etc. are propositions concerning objects a,b,c,d. These propositions can be true as well as false. 
Expressions F(x), G(x,y), H(x\,- • • ,xn), I(x,x), etc. are predicates, tha t is, functions of arguments belonging t o a domain (field) TI. Further, formula \/xF(x) is a proposition being true if F(x) is true for each element of field TI and being false in the opposite čase. Formula 3xF(x) is a proposition being true if there exi­sts an element of field TI for which F(x) is true and being false in the opposite čase. F{x) can also be a propositional formula of predicates. 
A system of axioms for the predicate calculus is obtained, if to the axiom system I, II, ... , V of subsection 3.2 the following group of axioms is added: 
VI.	 Predicate axioms 
1) VxF(x)^F(y), 
2) F(y) -»• 3xF(x) 



3.6	 Inferential Axioms of the Predicate Calculus 
The question is how the inferential axioms of the predicate calculus can impact the construction of axioms in the informational calculus. In which respect could the informational entities "for ali" and "exists", which belong to quantifiers V and 3, respectively, be important at ali within the infor­mational calculus? So, let us make merely a short overview to the matter which is treated in detail in [5]. 
In principle, the inference philosophy of the predicate calculus does not differ essentially from tha t of the propositional calculus although the si­tuation with quantifiers brings differences which must be considered with special čare. As within the propositional calculus, the rules of substitu­tion and inference keep their central roles in the predicate calculus too. To make the difference clear, let us introduce the operator i (for 're­place') instead of S (for 'substitute'). 
In a predicate formula 21 (formula with predica­tes), a proposition A or a predicate F(-• •) can be replaced (substituted) by formula *S. In this čase, the substitution is marked by 
i*(2l) or iJfcf^Sl ) 

respectively, where A is a variable proposition, F a variable predicate of n variables; formula ®(*i) • • • jtfra) includes among their free variables specially marked variables t\,---,tn, the number of which is equal to the number of variables of predicate F, tha t is, n. 
Rules connected with quantifiers are the follo­wing: 
1.	
 / / *B —s- 2l(ar) is a true formula and ?8 does not include variable x, then *8 —> \fx 21(.T) is a true formula too. 

2.	
 If 2l(a?) —»• *B is a true formula and 93 does not include variable x, then 3x 2l(a;) —> 2$ is a true formula too. 


The basic inference scheme (rule of modus po­nens) of the predicate calculus remains 
21 21-+ » <B 

According to [3], various rules for the predicate calculus can be derived by means of the substitu­tion and inference rule considering the basic axi­oms. Such a scheme is, for instance, 
2i-»<8 

21-• <L 
21-+5BA <L 
or, in a general form, at given variables a, b, • • -, 6 

A.P. Zeleznikar 
21 -> B(o) 
21-» »(b ) 

2X -*• «B(t) 
2i^!B(a) A »(b)A---A»(e ) 


Another inference scheme, with the universal quantifier, being important for a later considera­tion could be 
21 -> «8(o) 21 -*• Va; »(s ) 

where x must not appear in *B(a). The next infe­rence scheme which comes out is, for instance, 
21 -• (® -+ H) 21 -»• ( S ->• 2J) 21 -•• ( » -»• i l A 93) 

This scheme can be extended by a transition from a two-part conjunction to the value domain of a variable a, that is, 
21 -> (<8 -»• €(a)) »-*(»-» Va; (L(»)) 

where in the premise a; must not appear. Analogously, for the existential quantifier, the scheme 
<B(a) -»• 21 3a; »(s ) -»• 21 

can be derived. This scheme corresponds to the disjunction scheme 
2i^ C 
2tVQ3-^ C 

etc. The listed inference schemes will be used for comparison with schemes of informational calcu­lus. 
3.7	 Theorems of Deduction within the Predicate Calculus 
The basic theorem of deduction in the predicate calculus is similar to the theorem in the propo­sitional calculus. But there are several theorems concerning formulas with quantifiers. 
Deduction Theorems for Predicates. If for­mula 23 is derivable from formula 21, then formula 
21^2 3 

is derivable in the predicate calculus, that is, 
h2t^23 

Further, for formulas including universal and exi­stential quantifiers, there is 
r-VxF(x) -> 3xF{x), 
Vx\/yF(x,y) = Vy\/xF(x1y), 
3xVyF(x,y) -* Vy3xF(x,y), 
H Vx(F(x) -»• G(x)) -> (\/xF(x) -• VsG(a:)), 
1- Vz^Oc ) -• G(s)) -> {3xF(x) -f 3a:G(a:)), 
h Var(f(a) = G(z)) -»• (Va:F(a:) = Va?G(a:)), 
3xF(x)	 = VxF(x), 3xF(x) = VxF(x), 
3xF(x)	 = VxF(x), 3xF(x) = VxF(x) 
ivhere 21 = 23 represents the formula 
(21 -> 23) A (25 -• 21) 
D 
Proofs for the listed theorems can be found in [5]. 
4 Axioms within an Informational Theory 
4.1	 Introduction 
The question is how could the axioms of the pro­positional and predicate calculus be turned over to informational calculus. The main difference se­ems to be between the realm of truth in the tradi­tional logic and realm of information in the infor­mational logic. The informational realm is sub­stantially broader and within it the truth appe­ars only as a very particular čase. Instead to say that formulas in the traditional logic are true or false, in the informational logic formulas can in­form in one or another way, truly and falsely, par­ticular^ and universally, in parallel and serially, straightforwardly and circularly, algorithmically and spontaneously, programmingly and intelligen­tly, routinely and creatively, logically (consisten­tly) and controversially, etc. Informational logic becomes an active part of any informational sy­stem, the theoretical and the practical one. 
Informatica 19 (1995) 345-370 351 

4.2	 Axioms of the Informational Calculus 
Axioms of informational calculus can find a logi­cal support in axioms of propositional and pre­dicate calculus (Subsections 3.2 and 3.5, respec­tively). At the first glance, the axiom designer can behave in a withholding way, considering the traditional axioms as much as possible. But, the­reupon, when getting the appropriate experience, the designer of an informational theory can go his/her own way, considering the entirety of the possible informational realm. Within this gene­ral scope, the true as a particular situation can be replaced by the informational as an extreme attitude of the informationally possible. The di­scussion in this section will follow the traditional way of axiom construction as much as possible. In the next section the most general and infor­mationally open way will replace the traditional thinking. 
First, let us classify the informational axioms according to the tradition in the propositional and predicate calculus. This way of interpretation will give us the necessary feeling of difference and in­formational generality in respect to the usual un­derstanding of "logical" axioms. It will be possi­ble to recognize the essential difference which go­verns the informational realm in its universality in comparison to the classical logical (propositional, predicate) realm. It certainly does not mean that the particular axiom. situation in logical calculu­ses does not nt the informational principles—it fits them in a particular manner. 
4.2.1	 "Implication" Axioms of the Informational Calculus 
In čase of implicational axioms, we are concer­ned with two basic possibilities. At the begin­ning of the axiomatizing process we are concer­ned with the so-called informational phenomena­hsm for which we have to design particular axi­oms giving us the certainty of our initial steps into a general theory of the informational. This means, we have to explain in a formally consi­stent way the arising of initial axioms themselves. For instance, we must induce the primitive axioms of externalism, internalism and metaphysicalism, which constitute the very general axiom of infor­mational phenomenahsm. 
In Subsection 3.2 we listed the group of axioms of implication. In informational calculus, we have a broader definition of the so-called informatio­nal implication, discussed in [10]. The question is where to begin the process of axiomatization in in­formational calculus. We are confronted with the principal difference existing between the truth in logical calculuses and the informing in informati­onal calculus. 
In the traditional logic, propositions and predi­cates inform in a true or false manner. In the informational calculus, informational entities— precisely informational operands—inform and are informed. Thus, we can choose the operand— informational entity—as the point from which one can begin the process of axiomatization. We shall see how different initial axioms will become pos­sible and how they will be circularly interwea­ved. This axiomatic analysis will deepen the un­derstanding of the informational phenomenalism, that is, the phenomenalism of the informational entity. 
According to Subsection 3.2, axioms I. 1-3, in informational čase, the following is obtained: 
I. Axioms of informational implication 
1) a => (/3 =>• a), 
2) (a=^(a=> P)) =} • ( a = » P), 
3) (a = > p) => ((/? = * 7 ) = • (a 7)) 
where a and /3 are informational operands (enti­ties) and =>• represents the operator of informati­onal implication (in the most general and complex form [10]). 
Prior to the informational systemization of im­plication axioms, let us look at examples which bring to the surface the logical sense of the li­sted implication axioms within the informational realm. 
An axiom of the form (1.1) 
a = > ((a |=) = > a) (1) 

seems to be regular. It means that an informatio­nal entity a simply implies that entity a is implied by informing of the entity, tha t is, by a \=. Theo­retically, it seems to be impossible to oppose such an initial principle because the informational na­ture of an entity is circular in respect to itself and its informing. If introducing a' s informing in the A.P. Zeleznikar 
form 3a, the last axiom can be expressed in the form 
(Jo, = > a) (2) 
a 

This form of the axiom is more general then the preceding one since it presupposes also the phe­nomenon of informational internalism, tha t is, 
a =*• ((|= a) =>• a) (3) 

However, there is not an equivalence betvveen the first (1) and the third (3) axiom on one side and the second (2) axiom on the other side. 
A resulting system of axioms concerning infor­mational phenomenalism in the sense of the pro­positional axiom (1.1) in Subsection 3.2 is 
a =>•(/? =*• a) ; 
P € {« K [=a > a \= «i (a N; N ")} 

or informationally explicitly 
(I 
a=* pe 
a 
\ \ 

or simply and evidently 
a =*-a 
I 

In this formula, the essential difference between "informational operators" comma and semicolon must be distinguished (the alternative and the pa­rallel operator, respectively). 
An extension of the discussed axiom system is evidently the following: 
An example of this system is not only the axiom 
(a h ) = > ((h «) = > (« H) 
which fits the general axiomatic scheme (1.1), but also 
( a |=) = > ((|= a) =$• (a \= a)) 
if arbitrary items of alternative array (a , a (=, |= a, a (= a, ( a |=; |= a)) are chosen. It is to stress tha t cases of non-informing, that is, a h L^ a, a\L a, etc. are understood as particular cases of informing. 
Informational axiom (1.2) offers other signifi­cant formulas (basic informational implications) which are used, for instance, in informational mo­dus ponens and modus tollens. 
One of the most important informational axi­oms, following the scheme (1.2), seems to be 
(a=*(a=> (a |=))) = • (a = > (a [=)) 
which delivers the necessary conclusion a =>• (a |=) needed in various rules of inference. 
A resulting system of axioms concerning infor­mational phenomenalism in the sense of the pro­positional axiom (1.2) in Subsection 3.2 is 
(a => (a => /?)) ==> ( a = > /?); 
P G { a |=, |= a , a |= a , ( a |=; |= a) } 
or informationally explicitly 
/ / / \\\ 
h«, 
a => a =>• /?€ < a |= a, • 
V l l HMJ J l 
a 
/ M 
V v 
One can imagine what happens if the first /? and the second /3 in the last formula are chosen as di­fferent entities which, in the last scheme, is possi­ble in several ways. Appearances of /3 within the alternative scheme (set) are legal independently of the randomly chosen element in each concrete čase. It is to mention that some alternative infor­mational schemes concerning the axiomatic atti­tude will also be discussed in Subsection 4.2.2. 
This particular axiomatic situation can now be transformed simply and evidently into a more ge­neral axiomatic scheme of the form 
Informatica 19 (1995) 345-370 35 3 
a a 
\ \ 

l /ah W 
|=a , 
a => a\= a, 
\ \\=<*JJJ 
The most general extension of the discussed axiom system could evidently be the follovving: 
A characteristic and progressively diverse form of the last axiom system would be, for example, 
( a = • ((a |=) =* (1= «))) = » 
((a |= a ) = > ( a h ; N a)) 
which shows a phenomenalistic sequence exten­ding in a straight implicative manner from infor­mational entity a over its externalism a |= and internalism \= a to its metaphysicalism a (= a and phenomenalism a |=; |= a. 
Axiomatic scheme (1.3) delivers a very funda­mental property of informing of two entities con­cerning the third entity. Such axiom is, for in­stance, . 
( a = * ( a h) ) = • 
l«N)^(N«) ) = (a = * (\= a))) 
or also the pair of axioms 
((a h) = » (h «)) =* (((h «)=^( 4 «)) =* a H => (« N «))); 
((a \=) = • (\= a)) => 
(lh)^(«hh)) 
(a hh «))) 


The general scheme for the implicative rule (1.3) becomes 
(«=•/?)= » ((/3 => 7 ) => (« = * 7)); /3,7 G {a |=, |= a, a F a , ( a F ; F a) } 
etc , in the sense of the previous examples (I.l) and (1.2). Thus, the informational version of the last svstem becomes 
/ / pa , a = ^ /?G< a p a, ^ 
C« NA \ H,N«JJy/ 

// / 
/3es 	7 € < 

\\\ 
/ 	/ \\\ 
P a, 7 G <a P a, 
fct)J

V 

The most general axiomatic version of the last svstem would be 
((a,  \  (a ,  \  
a  p ,  a  P ,  
h«, a p a,  =>•  p a, a p a,  =>•  
(N4  
\  /a ,  \ \  
a  p ,  
P  a,  P  a,  

= •
a p a, a p a, 
((<*, \ 
a |=, a |=, P a, P a, a (= a, =>• o; (= a, 
It h; )J (a\ 
=;)
a 

Evidently, from this axiomatic scheme, a "com­plete" implicative circular formula proceeds: 
A.P. Zeleznikar 

{a =>. (a p)) = • 
(((P a )	 = > (a p a)) (a|=;|=a)=>a) ) 
This formula is in no way an impossible specula­tion since ali operands are only different pheno­menal forms of one and the same informational entity a (with exception of the first and the last operand which are equal). 
4.2.2	 Another Form of "Implication" Axioms of the Informational Calculus 
Informational calculus bases on the informing of entities and not solely on the logical truth". As the reader can observe, the discussed propositi­onal and predicate axioms are always identically true logical formulas. We can show how other in­formational axioms which do not base on proposi­tional and predicate axioms can be derived intui­tiven, trivially and formally in the same manner as the preceding informational axioms. 
So let us take the syntactically rearranged axiom (I.l) in the form 
(a=>/3) a 

instead of a ==>• (/3 ==>• a). Propositional for­mula (A —• B) —> A is not an identically true formula and its value is A. Axiom (I.l) is called the axiom of the conseguent determination. What could the rearranged axiom mean at ali? Let us check its meaning by some basic examples. 
Let us look the semantic difference between for­mulas 
a (a |=) =$• a ) and («N» 
, a 	a 

If the first formula says that any informational entity a implies its extemalistic informing a \= as a reason of itself, the second formula stresses that any informational entity a is implied by an implication in which entity a implies its externa­hstic informing a \=. The reader will agree that it is practically impossible to argue against this ar­gument ation. In informational cases we do stric­tly distinguish between circular and serially non-circular cases. As soon as a |= appears within the cycle a ==>• ((a p ) =>• a), the transition a =L• (a p ) represents a part of the "whole hi­story" of the cycle and, therefore, can or must be 
considered within the form (a =>• (a [=)) = > a. Thus, another substantial implication comes to the surface: 
(a = • ((a |=) = • a)) = > ((a = • ( a |=)) = > a) 

Possibly, this fact becomes evident by the formula 
( a =>• 3a) => a 

where 3 a expresses the entire phenomenalistic na­ture of a's informing, that is, its hermeneutics (a kind of regular interpretation of a's history) which considers not only the history of both appe­aring components a and 3 a , but also transition a =>-3a (or 3a ==>• a in the first čase). 
4.2.3	 "Conjunction" Axioms of the Informational Calculus 
Let us draw an informational parallel to the pro­positional conjunction axioms (II. 1-3) in Subsec­tion 3.2. 
Wha t could be the conjunction in an informa­tional sense? How could it be generalized? 
The logical "and", represented by operator A, means also "and simultaneously" or "in paral­lel". Informational operator of parallelism is ||= or, commonly, semicolon ';'• Thus, for the first axiom of conjunction (II.l) there is, informatio­nally, 
(a\\=/3)=>a or (f\=><* ­

and for the second axiom of conjunction (II.2), 
(a |t= /3) = > /3 or ("A = • /3 

For the third axiom of conjunction (II.3.) there is 
(«=>/?)= * ((a = > 7 ) =j> (a = • (/3 |h 7))) 

or, in a common informational form, 
( a = > /3) = • L = • 7 ) = * L = • f #J ) 

The sense of parallelism axioms is significant in cases of the so-called informational decomposition (see Section 7). Two characteristic cases are, for example, 
4.2.4	 "Disjunction" Axiom s of the Informational Calculus 
Disjunction axioms (III. 1-3) in Subsection 3.2 introduce the meaning of informational formula «i> «2? •. • • j an where commas are used instead of semicolons. What does, in an informational for­mula, a comma mean at ali? 
Informationally, propositional axiom (III. 1) can be interpreted as 
a=>(a,p) 

Instead of propositional formula A V B there is in­formationally, simply a, (3 where the last formula just lists two alternatives which are a and /3. Al­ternatives are informational entities which can be chosen by an informational system. 
From which point of the philosophy could the discussed alternativeness come from? In an exter­nalistic čase, a |=, intuitively, a presumption a \= /3 exists, otherwise the externalism of a would not perform meaningfully. One can agree that from the process represented by formula a \= /3 ope­rands a and /3 can be listed. Thus, 
(a\=P)=>(a,P) and (a |=) = * (a,/3) 

Through this demonstration the step towards the 
"disjunction" axiom in informational calculus be­
comes evident. 
According to propositional axiom (III.2), there is 
/3=*(a,/3 ) 

which is another form of the first disjunction in­
formational axiom. It is not meant that the list 
a, /3 is ordered. 
Finally, according to propositional axiom (III.3), we have 
( a = > 7) =>• ((/? =>. 7) = > (a,/3 =$> 7)) 

In the	 last axiom we have to explain formula 
a, P =>• 7 additionally; the meaning is the fol­lowing: 
(a,/3=^ 7 )-(^^ ) and 
(a,p)^{a,P} 

Thus, a list a, /3 connected with an operator in a 
formula results into a parallel system. In general, 
•K/s h T)-(L[:? ) ™i 
K//N)-(;f) 

This short discussion rounds up the possibilities of construction to the propositional disjunctive axi­oms parallel informational axioms. 
4.2.5	 "Equivalence" Axioms of the Informational Calculus 
The reader might observe that we have never de­fined a rigorous informational formula by which informational operator of implication, => , would be defined once for ali. In [10], only verbal possibi­lities of the informational contents of implication have been listed. On the other hand, propositio­nal implication, —•, is defined by a kind of logical table once for ali. Does it mean that the contents (not a rough definition itself) of informational im­plication changes (arises) from čase to čase? The answer might be the following: operator =>• is a regular informational operator which, from čase to čase, can (must) be particularized and univer­salized, according to the involved operands. So, it must be interpreted only to the sufficient in­formational extent. The reader can imagine how complex and never ending interpretation of infor­mational implication would proceed from the ba­sis of its verbal (dictionary-like) determinations. 
This fact does not restrict the discussion of informational equivalence in which informational implication plays an essential role. Thus, we can put the question of informational equivalence in parallel to the existing notion of propositional cal­culus. 
According to the axiom group (IV. 1-3) in Sub­section 3.2, the adequate informational ,axioms concerning informational equivalence are the fol­lowing: 
( a «=> /3) = > ( a =$> /3); 
( a ^=> ^ ) = • (j9 = > a); 
(a = > /3) => ((/? => a ) = > (a <=?> /3)) 

Informational operator of equivalence, •*=>•, can be defined dependently on informational implica­tion, that is, according to deduction theorems of predicate calculus in Subsection 3.7. In this sense, A.P. Zeleznikar 
(a <=> /3) ^ De f ((a = • /3; /3 ==• a)) 

Thus, in each particular čase, when particulari­zing operator •$=>, this particularization depends on the particularization of implicative operator 
4.2.6	 "Negation" Axioms of the Informational Calculus 
The purpose of the negation of a statement in pro­positional calculus is to obtain the true negated statement, that is, the true otherwise false state­ment. In informational calculus, an entity informs or can be informed in a certain manner, but not in another one. This last situation can be discussed in the framework of an entity's non-informing. 
Thus, an informational equivalent to the propo­sitional situation A is, for example, a \L and \fi a which reads a does not inform (in a certain man­ner) and a is not informed (in a certain manner), respectivelv. Additionallv, operator \fi represents only a particular čase of operator |=. 
For a double negation A it is possible to distin­guish different cases of double non-informing, for instance, 
(aftfc &{<*&>, (fet a) fež; \L (fcfe a ) 
Considering the axiom group of negation (V.1-3) in propositional calculus (Subsection 3.2), there is, informationally, 
(a = * /3) = • ((/3 ^ /3) = » (a ft \fi a)); 
However, to these axioms, characteristic informa­tional axioms concerning the phenomenon of non-informing can be adopted. For example, 
(a = • (a h)) =^(a=^(a ^)) ; 
(a = » (|= a)) = • ( a =*• ( ^ a)); 
( a = > ( a |= a)) = > ( a =>• (a \L a)); 
(a =^ (a |=; \= a)) ^(a^(a^; ^ a)) 
etc. concern operator particularization in the pro­cesses of decomposition. 
4.2.7	 "Predicate" Axioms and Theorems of the Informational Calculus 
The universal and existential quantifi.er do not play a significant role in the informational calcu­lus. They are rather very uncommon entities wi­thin the informational realm. For instance, ope­rator |= ve* reads informs (is informed) for ali a 's and formulas 
 o r a
" |=for_all_a \=Va] |=for_all_a « Or \=Va &; « Nor-all_a « Or. a\=yaOL\ O Nor-all^a! |=for^,U_a « Or O \=~ia\ \=\/a a 

are very special cases since it is not clear to which informational realm the 'ali' could refer. A simi­lar, problematic situation occurs in the čase of the existential quantifier where formulas 
 o r a
a |=exist_a t=3« ; Kxist ^ a or \=3aa; ot |=exist_a « or a |=3 a a; C F=exist_aj r=existja & Or OL |=3a!) p1 3a ® 

Analogously to the predicate axiom group (VI. 1­2) in Subsection 3.5, the adequate informational axioms could take the form 
V(^)=> (« Na« Y>(«)) 

The reader can imagine how the operator attri­bute for ali a 's- in an informational realm causes that the informational function (p is informatio­nally impacted by (3 which is within the realm of aH a, etc. On the other hand, informational function </?(/?) (as determined in [9]) implies the existence of function ip being dependent on /3, etc. 
In parallel to the deduction theorems for pre­dicates in Subsection 3.7 it is possible to derive "similar" informational theorems for informatio­nal entities. 
Decomposition Theorems of Informational Entities. If formula (3 is informationallv deriva­ble from formula a, then formula 
is derivable in the informational calculus, that is, 
-»(a=>/3 ) 

The theorems of predicate calculus in Subsec­tion 3.7 become, informationallu, 
Informatica 19 (1995) 345-370 357 
-*((a Nv« tp(a)) ==> (a \=3a <p{a))); ( a |=v«(/? h v? Y>(<*,/?))) «=> (P hv,e (a |= v« Y>(<*,/?))); (a\=3a(/3 \=v0<p(a,/3)])<=> ((3 \=vp(a\=3a v(a,/3))); -»((a Nvcv (v(a ) =>• ^(a))) => ((a |= V a </>(")) = > ( a Nv« ^(«)))); -»((a |=Va (v(a ) =>• V»(a))) => ((a 1=3« </?(«)) = > (a ^ 3 a ^(«)))); -»((a hv a (<p(a) <=» ^(a))) = > ((a ^v a Y>(«)) ^=^ (« hv a ^(")))); (a t=3a <p(a)) «=> (a ^ a (y?(a) \L; ^ (p(a))); 
(a|^ a(^(a)^^(«))); 
(ah»W«)^;^W)); («|=3a (v(a) b^; b^ y(«))) <=>• O Nvc Y»(«)) 

D 
These theorems can cause a retrograde and logi­cally more critical understanding of the predicate theorems listed in Subsection 3.7. 
4.3	 The Variety of Informational Axioms of Inference 
One of the basic questions concerning the infe­rence schemes in different calculuses is in which manner inference rules are constructed and accep­ted by scientific communities. Obviously, the way to various inference rules leads through the un­derstanding of the basic axioms treated in the preceding subsections. An inference rule is no­thing else than a means for the derivation (de­duction) process where it is used for a construc­tive transformation of theory-legal formulas into new formulas. The next step concerning axioms and inference rules is the introduction of the so-called replacement rules by which formulas can be transformed automatically, e.g. in a machine-like fashion. 
Informational rules of inference function like axioms and only by them it is permitted to con­clude in a theory-consistent manner. E.g., rules of substitution and modus ponens belong to the most obvious and widely accepted rules for for­mula transformation within a deduction process. On the other hand, informational inference rules can cover informationalb/ unlimited possibilities and they can come into existence together with specific and particular informational problems. 
The so-called modi informationis [6] can.be gro­uped and expressed in several possible ways. We have the following possibilities: 
1. Modus ponens is the most obvious inference rule in mathematics in the realm of truth. Infor­mational modus ponens follows the general (phe­nomenalistic) principle of informing where the mathematical to be true is replaced by the infor­mational to inform. This does not mean that the true is excluded, it only appears as a particular čase of the informational. In this sense, informa­tional modus ponens as an inference rule retains the basic logical form which is 
a;(a=> /?) 

and where the implicational operator =>• has a broadened meaning in comparison with the logical implication operator —>, determined, for example, by a logical table. Implicit informational operator of parallelism ';' which simultaneously performs as a formula separator replaces the logical opera­tor A (also, a comma in the traditional inference rule). 
Particularizations of modus ponens can appear as special rules called, for example, modus proce­dendi, modus operandi and others. 
2. Modus tollens is a rule of a reverse inferring in respect to the implication formula of its pre­mise. There is a truly substantial difference be­tween the mathematical and informational modus tollens. In the domain of truth, the falsity is the only essential counterpart to the truth and, so, the operator of negation of a formula becomes truly essential. Through the introduction of negation it becomes possible to express the falsity of a sta­tement as its truth (e.g., identically true falsity). 
In informational modus tollens the negation is replaced by the operator of noninforming (in a certain way), denoted by \L. The reader can now feel the dramatic difference existing betvreen a static state of negation and the dynamic state of noninforming concerning a logical and an in­formational entity, respectively. Thus, the infor­mational modus tollens in comparison with the logical one keeps the form 
A.P. Zeleznikar 
(a = » /3); (/3 fr & /3) a\L;\La 

with already mentioned differences concerning the modus ponens. 
3. Modus rectus is not a mathematical inference rule and its origin can be searched in the analysis of the Latin speech. The aim of this rule is the filtering-out (detaching) of the intention marked by iintention(Q;)> hidden in entity represented by a through the informing of a, e.g. to an entity represented by /3. It is understood that this in­tention is an informational function of a which is reflected in /3 to which a informs intentionally (or in an intending manner, that is, 'intendingly'). In this sense, apossibility of informational modus rectus becomes 
O? ((<* Nintend P) => *intention( Op) (iintention(Q!) N i h MntentionC«)) C /3 

One can certainly construct 'intentional' rules where the intention or an intention-like entity (functional operand) performs in a certain man­ner, so, its detachment becomes possible. 
4. Modus obliquus belongs to the most conten­tious inference rules because it usually proceeds from an absurd situation where the inference from a contradictory situation suggest to use the rule for an achievement of the logically consistent re­sult. From a deceitful situation just the absurd inference should help to reach a firm conclusion. In fact, modus obliquus belongs to the so-called discursive informing where through a discourse (as communication, informing, reasoning) by ali possible logical tricks, including absurd, contra­diction, controversial informing, a valuable con­clusion should be detached. In this sense, modus obliquus becomes a discursive filter delivering a useful result. In this sense, one of its possible schemes could be 
»absurd(a ) C <X\ («absurd(o ) N h <*absurd(oQ) = » P) (aabsurd(o;) N i \L «absurd(o)) C /3 

where aabsurd(«) marks an absurd component of entity a hidden in a which phenomenalistically informs /3, so that its informational phenomena­lism can be identified in /3 as the observer. 
The presented example of modus obliquus shows how other inference schemes would be pos­sible and how, by further decomposition of the rule and its components, more complex and se­mantically concrete inferential schemes could be developed. 
5. Modus procedendi is a mood by which a goal in­formational entity flgoal(a) will be detached from the informing entity a. An example of modus procedendi is 
(flgoal Cgoal a) ; (a =^ g oa l flgoal) . " Fgoal (flgoal Fgoal <*) 
where implication operator =L• is particularized to =4> goai, causing a consequent goal-like infor­ming (operators Cgoal and hgoal)- The conclusion of the inference rule can certainly be extended, bringing other components of the goal structure to the parallel interpretative surface, for instance, 
« Fgoal (flgoal Fgoal Oi); ( « Ngoal flgoal) hgoal « ! flgoal h (^goa l h P goal)! (flgoal h ^goal ) h fl goal) 
To this goal system within entity a other inter­pretative formulas can be added. 
6. Modus operandi discovers the inside of an en­tity, its modus operandi. The inside structure of an entity a is its metaphysicalisrn in the form of informing 3a, counterinforming €a and infor­mational embedding <La, together with counterin­formational entity ca and embedding entity ta. Thus, the consequent inference rules are nothing else than circularly structured modi ponens, for example, 
<*;(<* =>3a) 3al(3a <L*) 
La;(<La =F> C a) . 
<§a ' (L«;(L* =^ea ) Ca i (Za ^ Oi) 
a 
An additional inference rule (modus operandi) co­uld be 
a; (a {t* C <L») C Ca) C <ta) C 3a) C a) 
Ca i *-a> C a , <~cn J a 
Informatica 19 (1995) 345-370 35 9 

7. Modus vivendi is certainly an informational čase, for instance, how to infer in an (extremely) critical situation. Modus vivendi does not neces­sarily consider an extremely logical intention, so it can extend from an uncertain situation, e.g., mo­dus possibilitatis, modus obliquus to modus po­nens, as the most certain, approved and standard rule of inference. 
Informational modus vivendi of an informatio­nal realm concerns inference in situations of life and surviving compromises happening under en­tity's environment, individual, populational and social circumstances [6]. 
Let the sensory information cra((3) of entity a observing entity (3 inform the a's metaphysicali­stic structure, marked by a hmetaPhysicaiistkally <*,• where the metaphysicalistic structure of a is so­mething represented by the circular formula 
a h P a h (<L« h (Cev h (<L« h (Ca h «))))) 

and other formulas belonging to the a's me­taphysicalistic gestalt [9] of length t = 6, Thus, \ P h (act(P) h (a hmetaphysicalisticaUy «)) 
The last formula is nothing else than an informa­tional decomposition of transition j3 \= a where the essential operator frames [9] in this formula are 
P h (<7a(P) h ia hmetaphysicalistically Oi) ^)J an d 
P h (<^a(P) h (a hmetaphysicalisticaUy « | )) [ 

These cases of transition (3 \= a can be interpreted in the form of the so-called operator composition /3 h/3 ° h « ai that is, 
P h/ ? ° haa(/3) ( a hmetaphysicalisticaUy « ) an d 
P |=/ 3 ° F;CTa(^)[=metaphysicalistically a ** 

By observing of (3 by a, the so-called behavio­ral information /?behavior(c*,/?) pr°duced by the a's metaphysicalistic structure, through its sen­sory information ca(/?) and, dependent on enviro­nment /?, has to be detached, where /?behavior(<*, P) is on the way to assure the survival of a depen­ding on environment al information /3 (in the sense of modus vivendi). Within this discourse, the fol­lowing rule of modus vivendi arises: 
/K(č)c(/?h«)); \ 
((VaiP) \= ( a |=metaphysicalistically «) ) ==> \ /?behavior(«,/?)) / Pbehavion^ j P ) 
Senseful other examples of informational modus vivendi can come to the surface in more concrete informational situations and attitudes. 
8. Modus possibilitatis roots in the philosophy of modal logic [1]. Instead of <>A which means possibly A (exactly, A is possibly true), we can introduce a |=poSsibly which reads a informs pos­sibly. A shortcut would be a§. Thus, a<0/3 is read a possibly informs (3 or also /3 is possibly in­formed by a. Sometimes, a |=<> /3 is an appropri­ate notation for a čase of possible informational transition. 
The question is, for instance, which are the pos­sibilities of ,an informational entity a's arising. How can a possibly arise? The possibihty of a's arising depends on the possible informing of a itself and on informing of an exterior entity, say /3, which could possibly impact a informationally. Thus, the basic situation is a double transition a, P |=<> a with the meaning 
WN»)=(^:: ) 


Let x<> mark a structure of informational possi­bilities, where 7TI,7T2, • • • ,irn C 7r<>. These com­ponents can be detached transitionalb/ by modus possibilitatis in the form 
(q,/?,a-$);((q,/?fc=0 a) = » TTp) 7ri,7r2, • • -,7rn C 7r<> 

Transitionally means, that the conclusion has to be unfolded in the sense of the operator C de­fmition [8]. In general, by informational modus possibilitatis it is possible to infer about possibi­lities of informing of interrelated entities. 
9. Modus necessitatis, also, roots in the philoso­phy of modal logic [1]. Instead of • A which means necessarily A (exactly, A is necessarily true), we can introduce a |=necessariiy which reads a informs necessarily. A shortcut would be aO. Thus, aOfl is read a necessarily informs /3 or also j3 is neces­sarily informed by a. Sometimes, o. |=D (3 is an A.P. Zeleznikar 
appropriate notation for a čase of possible infor­mational transition. In modal logic, the interplay of possibility and necessity becomes the essential point of formula derivation. Thus, in modal logic, a schema embodying the idea of that vohat is pos­sible is just what is not-necessarily-not is given by the formula 
<>A <-» -.D-.A (or DA) 

An informational transcription of this modal for­mula would be 
(« No) ^=^ (« &) \^n 

with the meaning 'a informs possibly' is informa­tionally equivalent to 'a does not inform, does not inform necessarily'. 
The question is again, \vhich are the necessities of an informational entity a's arising. How can a necessarily arise? The necessity of a's arising depends on the necessary informing of a itself and on informing of an exterior entity, say /3, which could necessarily impact a informationally. Thus, the basic situation is a double transition a, (3 \=n a with the meaning 
<«•>*">- (? N a) 

Let va mark a structure of informational necessi­ties, where v\, vi-, • • •, vn C ^n. These components can be detached transitionally by modus necessi­tatis in the form 
(<x,P,va);((a,/3 \=g a) = » ua) 

Transitionally means, that the conclusion has to be unfolded in the sense of the operator C defini­tion [8]. In general, by informational modus ne­cessitatis it is possible to infer about necessities of informing of interrelated entities. 
5 Ceneral Schemes of Informational Axioms 
Up to now we have discussed the nature of in­formational axioms rooting to some extent in the tradition of mathematical logic. This tradition has its foundation in the groups I-V of propo­sitional.axioms dealt with in Subsection 3.2 and the two predicate axioms in the group VI in Sub­section 3.5. The critical question is if these axi­oms can be generalized in a more radical way as they have been changed in an informational man­ner through a slight universalization of proposi­tional operators when replacing implication —•, conjunction A, disjunction V, equivalence = and negation ~ by informational implication =>•, pa­rallel operator ';' (semicolon), operand separator ',' (comma), informational equivalence <=>• and operator of a (certain) noninforming \fi, respecti­vely. In čase of predicates, the predicate expres­sions Va and 3a representing the application of the universal and existential quantifier, must be replaced by informational operators of the form |=Va and \=3a, respectively, where the concerned entity a had appeared in the operator subscript. At this occasion, the question of the informational nature of logical (predicate) quantifiers had come to the surface as a reflection which, possibb/, may demand a deepened (more critical) informational treatise. 
5.1	 Informing versus Informational Implication 
Implication belongs to the most significant tradi­tional logical concepts. In this respect, the role of implication is revealed or hidden practically in any logical axiom and inference rule. Implication is the basis of the traditional logical calculus. 
In informational calculus, the role of implica­tion can be generalized. By generalization, the operator of informational implication, => , is re­placed by the most general informational operator (joker), |=. In this čase, the concluding from one true situation to the other can be replaced by in­forming from one informing situation to the other. In the uttermost situation, this kind of the conclu­ding informing can be comprehended as a general type of inference in which the traditional if-then­ism (implicativeness, causality, consequentiality) is replaced by the most general informational im­pacting in the sense of informing (informational externalism) and observing (informational inter­nalism), tha t is, in a form of informational tran­sition occurring (happening) between the infor­ming entity (emitter) on one side and the obser­ving entity (receiver) on the other side. The in­formational transition between the informer and observer can be discussed in the form of an infor­Informatica 19 (1995) 345-370 36 1 
mational operator decomposition where one part of this decomposition belongs to the informer and the other part of decomposition to the observer. This kind of transitional decomposition between the informer and observer can be understood as the operator composition between two operands. 
Let us discuss the axiom groups I-V in Subsec­tion 3.2 and the group VI in Subsection 3.5 un­der the most general informational'circumstances. There is: 
I. General axioms of informing 
• 1) a h (/3 1=«); 
2) (a |= (a N/3)) N (« N 0); 
3) (a N č) N P N 7) N (« N 7» 
II.	 General axioms of parallelism 
1) (a;0)\=a; 

2) («;/?) N/?; 3) («Nč)N ((aN7)N(*N(/3;7)) 
III.	 General axioms of alternativeness 
1)	 ah«, A 
2) /3\=a,/3; 
3) (akP)\= 

((/3h7)N(«,/9N7)) 
IV.	 General axioms of equivalence 
1) (ah=/3)N(«N/?); 2) («N«H^N«) ; 3) (a|=/3)N(G0Na)N(«N=/?)) 
V.	 General axioms of noninforming 
1) (a\=(3)\= 

((^N;N/3)N(«N;N«)); 2) aM(aN;N«)N;N(«N;N«)) 3) ((a ^ ; ^a)h;N(aN;N«))N « 
VI.	 General functional axioms 
1) (aNvaV(a))Nv(/?) ; 

2) ^)N( « Na« Y>(")) 
These axioms can be commented in several ways. Axiom (1.1) is evidently a satisfactory one. It says tha t entity a has, in general, its informer /? and that this fact is informed by a itself. If one takes data S, (1.1) becomes 
Data informs as a fact and cannot be informed. Operator \fc is nothing else than a particular infor­ming, that is, non-informing. Comments to other axioms can become not only very challenging but 

also provocative, controversial and novel for the usual understanding. 
5.2	 Generalized Initial Informational Axioms 
The very initial informational axioms will always remain within the scope of the informationally possible (the informationally arising). Which co­uld be the most universal (initial, original) infor­mational axioms? To answer this question one must certainly consider the specific informatio­nal phenomenalism with its four basic phenome­nal forms which are the externalism, internalism, metaphysicalism and phenomenalism. These axi­oms have not the usual implicative form but the universally informing one. 
Some of the most general informational axioms are those concerning an informational entity's ba­sic phenomenalism. For instance, axiom 
« 1= ((« H) h <*) 

can be read in the following way: "Operand a informs in such a way tha t it is informed how 
does it inform." Axiomatic formula 
« N ((N ") 1= «) 

can be interpreted as: "Operand a informs in such a way tha t it is informed how it is being infor­med." Axiomatic formula 
a \= ((a \= a) |= a) 

can be read as: "Operand a informs in such a way tha t it is informed how does it inform and how is it informed within itself." At last, axiomatic formula 
« |= ((a |=; |= a ) (= a) 

can mean: "Operand a informs in such a way 
tha t it is informed how does it inform and how is 
it informed as such." 
The listed four general initial axioms, procee­ding from (1.1), are structured in a strictly obser­ver situation when the main operator (= directly follows the leftmost operand a. The other possi­bility would be a strictly informer situation when the main operators precedes the rightmost ope­rand. For this čase, the listed four general axioms become A.P. Zeleznikar 
(a \=	 (a h)) h «1 
(a h (N «)) h «; 
(a\=(a\= a)) (= a; (a\=(a\=;^a)) \= a 

The point of informer philosophy is that , vvithin the informational transition betvveen a as infor­mer and a as observer (an a-circular čase), the entire additional information [a |=, |= a , a (= a, and ( a |=; |= a)] , is on the informer side. 
5.3	 Generalized Inferential Informational Axioms 
In a similar way as the basic axioms, it is possible to generalize the well-known inferential axioms, e.g., modus ponens, modus tollens, modus rectus, etc. A list of such generalizations of informational inference axioms would be the following: 
GMP. (a;a\=P)\=P\ 
GMT. (aM;(^;N))N 
(«N;N«); 

GMR. (a; (a |=intend P) \= tintention(a))) 1= 
((iintention(o:) N N iintention(a)) C P)] 
( aabsurd(a) C a ;

GMO.	 \l= 
\(aabsurd(aO |=; 1= «absurd(a)) \= P J 
("absurd^) N; \fittabsurd(tt)) C /?J 

GMPr. (sgoa l C a; a |= 0goai) |= 
(« h (flgoai h ")); 
(a ; a |= J a ) |= 3 a ; 

\Ja', Ja r~ *<*/ F" ^-a! 
(SalČa \= Ca) (= <:„; 

GMOp. 
(, Ca i Ca F ^a) r^ *-a! 
(GalGa \= ta) \= ta] 
(ea;ea |= a) \= a 

(Mf3)c(/3\=a)); \ 
GMV. (MP) \= (a |= a)) |= |= 

\ A>ehavior(a,/?)) / 
Pbehaviorl, & i P ) j 

GMPo. ((a,^,7ro);((a,/3(=o a ) (= 7r<>)) |= 
(7ri,x2,-• -,7Tn C 7T<>); 

GMN. ((a,/8,i^);((a,/8|=D a ) |= va)) \= 
{v\,v2,---,un C fD) 

This list of inference-axiom formulas includes the modi 1-9, discussed merely in a superficial (con­ceptually possible) way in Subsection 4.3. The main operators between the premise and con­sequence of each rule can be particularized by the use of the so-called detachment operator which, from čase to čase, can be specifically decompo­sed, according to the transition between the pre­mise system (operand) and the consequence sy­stem (operand). Any inference transition from the premise to the conclusion can be understood to be a real informational discourse which under­lies the concept of regular informational arising (phenomenalism) in the scope of principles of cir­cularity, spontaneity, serialism, parallelism, etc. 
6 Axioms and Theorems of 
Decomposition within 
Informational Calculus 

The philosophy of informational decomposition of informational entities brings to the surface a substantial matter of deconstruction of entities which is not only semantic but causes seman­tic changes also through syntactic possibilities of formula-representing entities. Informational de­composition is a term which replaces the traditi­onal concepts of induction, deduction, abduction and the role of inference. Decomposition means that we are usually concerned with general entity concepts which have to be decomposed into gre­ater and greater detail. Decomposition also has the meaning of interpretation when formulas are added to the already informationally determined formula systems. Decomposition follows the in­formational possibilities of analysis with synthesis of formulas, by modeling, describing and realizing actively a certain informational system. 
6.1	 Axioms of the Informational Particularism and Universalism 
Particularizing and universalizing the existing in­formational formulas means to decompose (de­termine, concretize, supplement) them into gre­ater (more adequate, realistic) details. For exam­ple, formula a\= /3 can be decomposed in several ways, that is, in respect to operand a, operator f=, and operand (3. Such decompositions can be simple and also very complex. We have learned about two possibilities of operator decomposition using operator frames in Subsection 4.3, discus­sing the informational modus vivendi. 
Informatica 19 (1995) 345-370 36 3 

Particularization and universalization as a de-compositional procedure performs as a local transformation in the sense of analyzing and syn­thesizing informational operands and operators by bringing to the surface essential and possible informational details and building them into exi­sting formula structures. Both operands and ope­rators can be universalized (generalized) and par­ticularized (individualized), directly or interpre­tively in parallel, by operand decomposition and operator composition, that is in several possible ways. 
6.2	 Axiom s of Informationa l Serialism 
Informational serialism springs from the question how could an informational entity be decomposed in a serial form, to determine the serial structure (components) concerning the entitv. The basic axiom on this way is 
a 
" F P 

which is recursive regarding (3. Such a transitive recursiveness transits into a system of the form 
a «1 . an-i a \= «1' « i |= a2 ' ' ftn_i f= an ' "i,«2,---,o:n-i,an C a 

By the strict substitution from the beginning to the end (from the left to the right), one of the re­sulting axioms of informational serialism becomes 
a a p («i |= (a2 • • • |= (a„_i \= an) • • •))' 
tti,a2,-'-•,«„_!,«„ C a 

In fact, instead of a single decomposition formula, there can exist various decomposition formulas of length t — n, where operands a,- (i = 1,2, • • •, n). and to them belonging operators preserve their places and only the parentheses pairs are displa­ced in ali possible manners. 
On the other side, an informing entity a can form a serial composition with exterior compo­nents a2-, where ct; <f_ a. 
6.3	 Axiom s of Informationa l Parallelis m 
Informational parallelism emerges from the que­stion how could an informational entity be decom­posed in a parallel form, to determine the paral­lel structure (components) concerning the entity. The basic axiom in this direction is 
a o.i 
; r— ; «»•,«; c a 

a; c*i a,-; a,-p a j which is recursive regarding a; and aj . Such a transitive recursiveness transits, among other pos­sibilities, into a parallel inference system of the form 
a ui . «n-i a; a p «i ' «i;a i p a2 ' ' «n -i,an-i 1= On' a1 ,a2 ,-- -,«n-i,a n C a 
By such and such application of the listed infe­rential rules, one of the formulas for the parallel decomposition becomes 
a a;ai;ai p aJ; \ i, j = 1,2, ••••,nj 

In fact, instead of a single decomposition formula, there can exist various formulas for parallel de­composition of different lengths regarding the par­ticipating components a{ and transitions a; |= a j , which can be mixed in ali possible manners. If components a; are interior to a, there is a; C a . This condition [8] brings various possibilities for the decomposition-rule construction. 
On the other side, an informing entity a can form a parallel composition with exterior compo­nents ct{, where ai <f_ a. 
6.4	 Axioms of Informational Circularity 
The axiomatic view of informational circularity can stay within the framework of informational serialism. The only difference is that the first and the last operand in a serial formula are the same. 
The most important čase of informational cir­cularity seems to be the metaphysicalistic one in which the participating components belong to the main circular entity. This does not mean that cir­cularity cannot perform in an exterior way when information informed by an information source re­turns to it, adequately transformed by the exte­rior informational entities. Thus, evidently, 
' a 

a p (aX p (« 2 • • • |= (Oin-l \= (« n N «)) • • O)' ai,a2,---,an-i,an C a 
A.P. Zeleznikar 

could be one of other possible rules for serial cir­cular decomposition. In a parallel čase, the cano­nical circular decomposition 
a 
( a p ay\ a,-|= a; + i ; an |= a\ yi = l,2,---,n J 

could be one of the parallel characteristic cases. 
A particular and the most interesting čase ari­ses with the question of the circular causal in­forming. In this čase, several other rules can be constructed which even the participation of the components (entities, operands, formulas) wi­thin a causal loop. Causal problems of informing are studied fundamentalb/ and will be treated on some other places2 . 
6.5	 Decomposition and Composition Theorems of Informational Calculus 
According to deduction theorems of propositio­nal and predicate calculus (Subsection 3.4 and Subsection 3.7, respectively), it is possible to list some fundamental axioms and theorems of the informational calculus (see also Subsubsec­tion 4.2.7). But, decomposition theorems of infor­mational calculus can also become inferential, and new modi informationis can be derived according to informational axioms also beyond the discus­sed inferential rules in Subsection 4.3. In distinc­tion from inference rules in traditional logic, ali of which are axiomatically determined, the general informational theory enables a decompositional rise of inference rules, which become an essential source of the possibilities of informational arising of entities (informational operands and operators, the last ones by a procedure of decomposition). 
Decomposition is a much broader term than are, for instance, deduction, induction and ab­duction for it not only unites them but adds new possible principles of the spontaneously arising in the sense of modi informationis. 
Decomposition and Composition Theorem. 
If formula (3 is informational ly derivable from for­mulas a.\t a.i, • • •, o-at then 
Causal informing of serially and in parallel decompo­sed informational entities is studied by the author in the paper entitled "Causality of the Informational" which is in preparation and will appear soon. 
ai|=(«2|=(---(a»M)"0 ) 

is an informationally regular serial decomposition or composition formula and 
a i |= a2; 

is an informationallg regular parallel decomposi­tion or composition formula svstem. • 
By induction it is possible to show the following: if 
a1,a2,---,an _i,an -»/3 

then 
a1,a2,---:an~i -»(an \= P) 

where symbol -» is used for marking a derived formula within the informational calculus. 
A Decomposition Theorem Concerning In­formational Gestalt of a Serial Formula. Let 
a a
\= (« i 1= («2 (= (• • 'K-i h°n) " •))) be se­rial formula, where 0:1,02, •• -,an C a. It is to stress that the serial formula of length i = n can also have arbitrarilv distributed parentheses pa­irs and we mark it by o~-*(a, a\, a2, • • •, an). Let mark by'T^(a) the so-called informational gestalt assigned to a serial formula of length3 n, infor­ming from the left to the right, and let this gestalt be a parallel informational system of aH possible formulas of length n. Then, 
cr~>(a,ai,a2,-- -,<*„) T^(a ) 

uihere system T7\,(a) includes 
1 (2n\ 
n + 1 \n J 

3 The length t of an informational formula is always de­termined by the number of the occurring binary informa­tional operators of the type \=. In the debated čase, the number of occurring operands in the serial formula is L+1. Thus, the length of the formula is n, where there are n binary operators and n + 1 operands. 
formulas of length n. • 
This theorem will be proved on some other plače4 . 
7 A Preliminary Catalogue of 
Informational Rules of 
Decomposition 

Rules which can be used in a decomposition (com­position) process can be listed by a short catalo­gue. From these rules more complex and particu­larized rules can be constructed being adapted to a concrete (semantic, interpretative, arising) si­tuation (possible deconstruction). The presented rule catalogue has to be considered as a prelimi­nary one. 
7.1 Basic Phenomenalistic Rules 
An informational entity a can be decomposed in four basic forms, which are externalistic, interna­listic, metaphysicalistic and phenomenalistic, re­spectively, that is, 
a a a a a |=' |= a' a|=a ' a |=; |= a 

This initial system of transformation (decompo­sition) rules offers the necessary alternatives by which the informational scope (interpretation) of an informational entity is extended. 
An essential comment to the basic phenomena­listic rules concerns operand a which represents any informational entity in the following sense: if a appears in a rule several times, then each appe­arance could be comprehended as another entitv, say, /3, 7, etc. Such cases can become informati­onally legal in several decomposition processes of a's decomposition. Thus, also, 
a  a  a  a  a  
J^]  M ;  ^M ;  JW^]  JWi]  
a  a  a  

ah;M; /^N;N/3; TRFi etc , derived from basic phenomenalistic rules, are legal and senseful. Another problem is the so-called mutual independence of axioms which may not concern the catalogued informational rules at ali. 
4 Gestalts as informational entities are examined exhau­stively by the author within the study entitled "Informa­tional Frames and Gestalts" which is in preparation and will be issued soon. 
7.2	 A Formal Extension of the Basic Phenomenalistic Rules 
The basic transformation rules in the preceding subsection can be systematically permuted regar­ding to the externalistic, internalistic, metaphysi­calistic and phenomenalistic čase, respectively. There is 
a\= a \= a\= a \= a ' |=a' a \= a' a|=;|=a' (= a (= a \= a \= a a ' a\=' a \= a' a|=;|=a' a (= a a \= a a\= a a \= a a ' ot\= ' |= a ' a |=; |= a' a|=;|= a a|=;|= a a |=; |= a a|=;|=0 ! a a \= |= a a (= a 
By these rules, premises can be replaced by the corresponding conclusions of the rules. As one may observe, the rules 
a |= (= a a \= a a|=;|= a 
a a a a 

act in an reductionistic way (lowering the degree of a formula complexity). 
7.3	 Rules of Informational Parallelism 
Rules of parallel decomposition of an informati­onal entity root in the axioms of conjunction as particular cases of informational parallelism. It is to stress that the listed rules are in no way in­formationally independent because the aim of the rule catalogue is to show the reader various pos­sibiUties of informational decomposition. 
Parallel decomposition covers the view of the possibiUties of parallel interpretation of an infor­mational čase. The rules guarantee an introduc­tion of parallel cases which explain the respective čase into more and more details and in additio­nal manners. In this way, the informational sy­stem representing an entity grows and becomes more and more cofnplex, that is, informationaUy interweaved by its parallel structure. 
7.3.1	 Rules of Interior Parallel Decomposition 
Although the basic rule of an interior paraUel de­composition of an entity a is only one, that is, 
A.P. Zeleznikar 
a 
a; a 


this rule can be "multipUed" by considering the rules from Subsection 7.2. 
7.3.2	 Rules of Exterior Parallel Decomposition 
Which formulas can be understood as a paraUel performing of their components? An evident in­formational rule resulting from the basic axioma­tic philosophy is, for instance, 
(?) 


This mean that the rule of interior paraUel decom­
ot

position of entity a, which is , originates from 
a; a 

the initial metaphysicalistic nature of the entity, that is, 
a |= a 
a

Thus, rule is a consequence of impUcation 
a; a 
( a a \= a\ a \a |= a' a;ct J a\a 

7.4	 Rules of Serial Decomposition 
If decomposition of an entity a means an interior deconstruction (interpretation, analysis, synthe­sis) of a's informational structure, the appearing entities in a decomposition formula are informa­tional components of entity a. In the rules of the so-caUed canonical5 serial decomposition of a, 
5As mentioned in a footnote before, the canonical and non-canonical decomposition of informational entities (operands) will be discussed on some other plače, that is, in Informational Frames and Gestalts of the author, where concepts of canonical and non-canonical formulas will be presented in detail. 
a 
a \= («1 N («2 1=(-" (ttn-l |= «n) • • 0)) ' 
a 
( a ^ ai ) ^ (a2 \= (•••(«„_ ! |= an ) •••))' 
a (•••((a (= ai ) \=a2) •••)]= (a„_ i (= a n ) ' a ((• • -((a ^ ai ) |= a2 ) • • •) |= «™-i) \= an 

where 
aa,a2,---,an C a 

This means: as soon as an entity a,, i = 1,2, • • •, n is introduced (comes to the surface, ari­ses, appears) in the serial structure of a' s decom­position, it is internalized through the transition formula ct; Ca [8]. Other forms of serial decom­position are the so-called non-canonical formulas of length n, for instance formula 
a 
(a |= (ax |= a2)) N («3 N (• • • («n-i N "n) • • •)) 

in which the parenthesis pairs are non-canonically distributed. 
7.5 Rules of Circular Decomposition 
Rules of circular decomposition consider serial, metaphysicalistic, causal, and parallel decompo­sition. 
7.5.1	 Rules of Circular Serial Decomposition 
A primitive, non-trivial rule of circular serial de­composition of entity a concerns a' s informing 3a, that is, 
a a a h C3a \= a) a n (a \=?a)\=a 

Rules for a general circular serial decomposition are characterized by conclusion formulas in which the first and the last entity are marked equally. Thus, the n+ 1 canonically-serial circular formulas are 
a 
a \= (ai \=(a2 h (• •

•KN)-)))' 
a 
( a |= ai ) h («2 h (• ••(an |= a ) ••.•)) 

a 
(•••((« h«i ) N«2)-••)hKN« ) 

a 
((• ••((«!= ai ) |= a2 ) • • •) 1= "« ) h « 

where 
ai,a2, • • -,Q„ C a 

On the other side, there are stili 
1 /2 n + 2 \ . . 

non-canonical serial circular formulas of length n + 1. 
7.5.2	 Rules of Serial Metaphysicalistic Decomposition 
Metaphysicalistic decomposition considers stan­dard interior components of an entity a which are the following: informing 3 Q , counterinforming <La, counterinformational entity ca, informational em­bedding <La, and informational embedding entity ta. Six canonical forward-cycle (right-loop) me­taphysicalistic rules are: 
a 
a \= C3a N (<L« N (Ca N (€a N ( ^ N «))))) ? 
a 
(a |= j a ) N ( ^ N (C* N (<L* N (ea N «)))); 
a 
((a |= Ja ) N ^a) H (Ca 1= («a N (Ca N «))) '' a 
(((a |= 3 a) |= <k) h Ca) N («a N ( ^ h «)) ' 
a 
((((a |= J a ) |= €a) \= ca) \= <La) h (ea h a)5 a 
(((((a h	 ?a) h *a) N Ca) |= <La) \= ta) \= a 

To these 6 rules 126 non-canonical forward-cycle metaphysicalistic rules can be added. 
Six canonical backward-cycle (left-loop) me­taphysicalistic rules are: 
a 

a \= (ea h («a N (Ca N («L« 1= P « N a))))); 
a: (a |= eQ) |= (€a |= (ca M^ h (% h «))))! a ((a |= e«) N <S«) h (Ca h (<L« N P a h «))); a (((a |= e„) h <S«) h Ca) |= ( ^ 1= P « N «)); o; ((((a |= ea) \= <L«) h ca) |= <L„) |= p a |= a); a (((((a ^ Za) \=<La)\= Ca) \= (ta) h 3« ) \= « 
To these 6 rules 126 non-canonical backward-cycle metaphysicalistic rules can be added. 
7.5.3	 Rules of Circular Serial Causal Decomposition 
Causal decomposition of an entity (operand) can be serial and serially circular. In the first čase, the difference between the ordinary serial and se­rially causal is only in the nature of informational operators. In the causal čase, they must perform in a causal manner, possessing the property of ca­usality. 
In a circular serial čase of causal decomposi­tion of an entity, the circularly involved entities (components of the decomposed entity) become equally entitled in respect to the cycle. This can cause the arising of components parallel loops and their gestalts. The consequence can be an enor­mous rise of the number of parallel cycles6 . 
7.5.4	 Rules of Circular Parallel Decomposition 
An operand a can be circularly decomposed in a parallel way by the following rule: 
a 
( a |= ai ; \ 
« i 1= a2; 
Otn—\ f= O!JU 
\ an |= a J 

The čase of causal circularity is studied exhaustively in Causality of the Informational, prepared by the author. 
A.P. Zeleznikar 

Such a rule describes a circular graph as shown in Fig. 4. 
Figure 4: A circular graph determined by the parallel svstem of transitions (a \= a\;a\ (= «2! •" •; ctn-i \= Oin;an \= a). The zig-zag co­urse of connections (informational operators) be­tween the informing components a,- of a points to a spontaneoushj discursive kind of informing. 
As we see, the graph in Fig. 4 can be the so­urce of different circular interpretations, stret­ching from parallel circularity to serial causal cir­cularity. It can simply mean that a parallel cir­cularity implies ali sorts of circular serial and cir­cular causal informing with ali possible circular gestalts. Within this comprehension, parallel cir­cularity is the source of ali possible rules of serial circularity within the informational calculus. For instance, 
1 a |= ct\\ ^ 
Oi\ \= «25 
Oi-a—1 p &n'i 
\ an\= a J 

fa\= (a i |= (a2 |= (•••(«„ |= a ) •••)));\ 
(a\=a1)\=(ai\=(---(an [= a) • • •)); (• • -((a |= ai ) |= a2) • • •) |= (an \= a) ; \((-• •((« N ai ) 1= a 2 ) • • •) |= a n ) |= a J 
is an example of the rule by which from a circular parallel system of n +1 simple transitional formu­las an adequate canonical circular serial system 
(gestalt) of n + 1 formulas is obtained. Certainly, such a parallel system can cause also the arising of a non-canonical gestalt of circular serial formulas. The most general rule for the replacement of a circular system of parallel transitions, marked by 7pl(a, ai , • • •, a n ) into a parallel system of circular formulas, that is, gestalts, becomes, for instance, 
7rll(a,ai, •••,an) 
rjfca^rg-1^,-); 
\ i,j = 1,2, •••, n / 
where r^" 1 (a ) is the gestalt of ali forward-cycle (subscript O) formulas of length i + 1 (i = 1,2, • • •, n). Similarly, T^"1 (a) is the gestalt of ali backward-cycle (subscript O) formulas of length i + 1 (i = 1,2, • • •, n). In the čase of the so-called causal circularity, components a\, a'2, • • •, an C a have the possibility to become informers and ob­servers in informational forward and backward lo­ops of lengths i+1, etc. So, T^faj) and r^" 1(«j ) are the corresponding gestalts. 
7.6	 Rules of Alternative Decomposition 
Rules of alternative decomposition embrace ali di­scussed rules in this section if the general informa­tional operator (joker) |= is replaced by the gene­ral alternative operator (alternative joker) =|. By this replacement, formulas' informing(ness) ope­rator becomes the informedness operator, that is, the operation of the Inform(s) is replaced by the operation of the Is/Are Jnformed. The deduction of alternative informing originates from two di­fferent sources. The first one comes from regular informing, where the basic rule of informational transition could be 
a \= P 
This rule determines nothing more than that for­mula a \= /3 can be read alternatively as /3 is in­formed by a, that is, (3 =| a. 
On the other hand, the alternative informing originates in the informational phenomenalism (Subsection 7.1), where 
a	 a a a 
=ja ' o; =)' a =| a ' =| a; a =| 
are the principled alternative rules. 
The basic alternative transformation rules can be systematically inverted regarding to the exter­nalistic, internalistic, metaphysicalistic and phe­nomenalistic čase, respectively. There is 
=\ a =\ a =) a =\ a a ' a=|' a=|a' =|a;a=|' 
a =\	 a =\ a =\ a =\ 
' i ' i ' i T' a =) « a =\ a =|a;a:= | a=| a a =| a a=| a a =| a a ' =j a ' a=|'=ja;a=| ' =| a; a =\ =\ a; a =| =| a; a =| =) a; a =) ' i ' i ' i 
a —\ a a =\ a =\ a 
By these alternative rules, premises can be repla­ced by the corresponding conclusions of the rules. As an example, the informational alternative­ness enables the follovving implication: 
/p=\a;  \  
^  ~|observe P J  
.  P  I—observe Oi,  
*>  '  Ql ^observing ^  P\  
P  ^impac t  &  
V  I  

The meanings of parallel formulas on the right side of ==> are not only informationally signifi­cant (logically instruetive) but also various in a semantic manner and are the following: 
-
 /3 is informed by a; 

-
 a is observed by (3; 


-f3 observes a; 
- a informs observingly /3; 
-/3 is mformationally impacted by a; 
- etc. 
8 Conclusion 
One of the aims of this paper is to show that the construetion of an informational calculus is possi­ble in a sufficiently (technically) rigorous way by a similar axiomatic approach known in the propo­sitional and predicate calculus. The difference is certainly obvious: informational calculus covers much wider realm of informational possibilities which may lie beyond the propositional and pre­dicate philosophy7 . What could such assumption mean at ali? 

The author is aware that the presented con­struction of the informational calculus is preli­minary and that stili many theoretical concepts must be developed prior to a satisfactory final re­sult. But it becomes also evident that a sufficien­tly developed theoretical approach builds up the fundament for something called the informational machine [11]. The main problem as seen by the author on this way is the so-called problem of in­formational arising which is the basic property of any informing entity. This basic property must be supported by the machine hardware and its opera­ting system, that is automatically, when an entity begins to inform as an informational entity of the machine. And, as we have experienced during the reading of the paper, a strong support to the in­formational arising of any kind lies in the decom­position phenomenality of entities informing seri­ally, in parallel, circularly, metaphysicalistically, and causally. 
A great deal of the problems pointed out alre­ady in this paper will be unfolded in the context of the ongoing studies concerning: 
1.	
 informational frames and gestalts, 

2.	
 informational transition of the form a \= /3, 

3.	
 causality of the informational, 

4.	
 understanding and interpretation, and 

5.	
 informational memory. 


By these studies, various cases of informational philosophy and formal theory will be presented and thematically rounded up with the aim to ena­ble the concept and implementation of the infor­mational machine. 
A good example of axiomatic informational possibili­ties is a derived axiom which considers the axioms of type 
1.3 in the mixed informing-implicative form, that is, 
This axiom is very useful and causes the so-called gestalt and causal development of formulas. 
References 
[1] B.F.	 Chellas: Modal Logic. An Introduc­tion. Cambridge University Press, Cam­bridge, 1988. 

[2] M.	 Heidegger: Sein und Zeit. Sechzehnte 
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1986. 

[3] D. Hilbert	 und P. Bernays: Grundlagen der Mathematik. Erster Band. Die Grundlagen der mathematischen Wissenschaften in Ein­zeldarstellungen, Band XL. Verlag von Julius Springer, Berlin, 1934. 
[4] G.	 Kwiatkowski (Ed.): Schiiler-Duden: Die Philosophie. Dudenverlag, Mannheim, 1985. 
[5] n.C .	 HOBHKOB: djieMeumu MameMamunecKOU AOZUKU. rOCVaapCTBeHHOe H3aaTeiTJbCTBO (|)H­3HKo-MaTeMaTirqecKOH jiHTepaTyp&i ($H3MaT­rH3), MocKBa, 1959. 
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