Informatica 19 (1995) 133-158 133 Principles of a Formal Axiomatic Structure of the Informational Anton P. Zeleznikar Volariceva ulica 8, 61111 Ljubljana, Slovenia a.p.zeleznikarOij s.si Keywords: axiom, circularity, derivation, functionalism, general informational theory, inclusiveness, inference rule, parallelism, propositional vs. informational, serialism Edited by: Vladimir Fomichov Received: June 6, 1994 Revised: August 27, 1994 Accepted: September 15, 1994 The article deals with some basic problems of a formal axiomatic structure pertaining to the phenomenalism of the informational. In this way, solid philosophical and formal foundations of an emerging informational science are set from the strict informational point of view [9]. A general informational theory conjoins the so-called object theory and its metatheory, in contrariness to a narrower mathematical theory, where the meta-mathematical theory serves as an exterior means for proving of the object theory. The principles of informational axioms are treated from the dualistic point of view conjoining the axioms of the object theory and inference axioms of the metatheory. Inference rules become regular informational formulas which arise inform at ionally as any other informational operands (entities). 1 Introduction Axiomatic structure1 of a general informational theory (GIT) is a problem per se, for it must be, for example, according to [9], self-contained in respect to the basic theory axioms on one side and the necessary initial inference rules (deduction, induction, abduction, modi of other possible inference) on the other side. Conception of GIT is certainly logistic [1] and formalistic [7], but not in the traditional mathematical sense. On contrary to the traditional mathematical theories, GIT can keep the inference rules within the theory itself while, in mathematics, deduction rules for example, remain outside the particular theory (e.g., in the so-called inferential metadomain, that is, me-tamathematics) as means by which a mathematician or machine can prove the correctness, logical consistency or non-contradictoriness of the theory. Mathematics is not more rigorous than histori-ology, but only narrower, because the existential foundations relevant for it lie within a narrower JThis paper is a private author's work and no part of it may be used, reproduced or translated in any manner whatsoever without written permission except in the case of brief quotations embodied in critical articles. range. (Heidegger [4], p. 195.) In mathematics, the metatheory by which an object theory is proved, lies outside of the object theory. It is mathematically unimaginable (uncommon) to join, for instance, an arithmetic theory (dealing with numbers) with the theory of deduction and induction, by which arithmetical theorems are proven and dealing with objects of logic of predicates. An object theory in mathematics is always meant as a narrower theory from which the metatheory is excluded. Axiomatization in the described (informational) sense is a necessary step towards a sufficiently strict theory which can be applied as a constructing or designing tool for particular informational machines and programs. It is a sort of informational formalization [7] by which a calculus is introduced. On each step of formalization, there is certainly possible a look into the real world when formulas are deformalized, that is, made less formal through their interpretation (translation) in a less formal or object language (natural, picture, voice, signal, process language, etc.) GIT is a theory of well-formed formulas of operands, operators, and parentheses pairs. It has a straightforward syntax where the formation 134 Informatica 19 (1995) 133-158 A.P. Zeleznikar of its formulas depends on several informational views, possibilities, and principles (methods), giving the theory the so-called informationally arising (emerging, generating) character. For instance, some of the principles of informational arising concern procedures of formula and formula system decomposition and composition. Decomposition means, for example, deconstruction [3] in the sense of a particular semantic and pragmatic analysis of a formal item (operand, operator, formula, formula system) in the form of additional arising, enlarging, changing or modifying. Several formal means have to be introduced before the axiomatization of the informational can begin. During our axiomatic discourse, we have the substantially different theoretical entities, the aim and purpose of which must be explained in a clear and definite manner. These entities are: 1. Definitions are a kind of preaxiomatic and pretheoretical determination entities which explain the introduced symbolism and symbolic structures (markers, variables, formulas, formula systems) used within the course of an informational theory advancing. Definitions are nothing else than transparent interpretations of formal symbolism for the reader in a natural language. They connect the emerging logistic and formalistic world ([1, 7]) with the natural one. They simultaneously enable the emerging of the formal theory world and its informational connection with a common (linguistic) individual consciousness. As such, definitions function like initiators beyond a particular axiomati-zed theory, linking the emerging formal world and the existing conscious world of the theoretically concluding mind. 2. Theory axioms are the essential origins of a theory obtained by an intuitive investigation of the basics by a theory setter (e.g., an expert of a scientific discipline). Informational axioms are formulas that commend themselves to general acceptance; they are informationally well-established and universally-concerned principles presenting the maxima of the possible, assignable degree of recognition. 3. Inference axioms are rules (laws) for deriving formulas from theory axioms and formulas obtained already through regular derivation. Inference axioms are the very initial rules for inferring, that is, for the drawing of conclusions from theory and inference axioms. Inference rules can be derived in the form of inference theorems, lemmas, consequences, etc. getting more complex and informationally in-terweaved inference rules. In this function, derived inference rules represent regular processes for drawing conclusions within an informational theory. 4. Theorems, lemmas, conclusions etc. are informationally derived theory formulas by means of inference axioms and inferred inference rules from axioms and already derived theorems, lemmas, conclusions, etc. They are "object-theoretical" (non-inferential) as well as "inferential". 5. Proofs of theorems, lemmas, and conclusions are procedures (informational processes) using inference axioms and generated rules with the aim to achieve certain results (in the form of theorems, lemmas, etc.) E.g., metamathematics can be understood to be a proof theory (D. Hilbert, see, for instance, [7]). In formally loose theories, the process of proving becomes an art instead of a formal procedure. 6. Analysis of theory axioms, theorems, and proofs occurs after the process of proving a theorem, lemma, etc. to see what could be improved, complemented, and added for the sake of a more complete and powerful theory. Thus, one can glance at induced axioms, derived theorems, and accomplished proofs. The enumerated theory entities (definitions, theory axioms, inference axioms, theorems, proofs, and analyses) constitute a spontaneous and circular discourse in the following sense: A. Construction of definitions and theory axioms is an informational approach by which the theory designer is getting his/her master for the emerging formal theory. The inference of further axioms and their notional improvement is on the way to the theory-axiomatic consolidation (fortifying, strengthening). PRINCIPLES OF A FORMAL AXIOMATIC Informática 19 (1995) 133-158 135 B. Construction of the inference axioms belongs to the functioning of the master and without them there would not be possible to deduce (prove) new axioms and the initial theorems, lemmas, and consequences. In case of an informational theory, inference axioms are parts of the particular informational theory and are not excluded from the object theory as it is the case in mathematics ([8], p. 30). Thus the entire informational master domain which governs the emerging of a theory uses definitions, theory axioms, and inference axioms for the mastering of the informational arising (development) of the theory. C. Construction of theorems lemmas, consequences, etc. brings to the surface the so-called university or teaching discourse. Theorems, lemmas, consequences, etc. can now be taught as a theory truth concerning the theory relevant entities. Derived theorems can be used in the same way as axioms together with the derived inference rules. But, new theorems have to be proved in a consequent manner, so that the teaching domain obtains the theory legacy. D. Construction of proofs can become a questionable task because someone wishes to prove a certain theorem which was constructed in advance, with the one's intention for some particular purpose, that is, intuitively. In this respect, proving of theorems can constitute the so-called histeric's discourse. In mathematics, the object domain (a theory) and the metadomain (metamathematics as a proof theory) are separated. A mathematician, proving his/her theory uses the me-tamathematical principles intuitively, for instance, in the mixed form of a natural and a formal mathematical language. E. Construction of analyses of the arisen theory constitutes the so-called analyst's discourse. Analysis governs the arising of cycles A, • • •, D and constitutes also the long cycle of a theory design, that is, A, ■ • •, E, A, • • •. The mentioned names (markers) of discourses have been invented by Lacan [14] and constitute a theoretical, cyclically and subcyclically structured discourse in its entirety. 2 Introducing Informational Operands and Operators Informational operands and operators are, together with parenthesis pairs, the basic entities of informational formulas. It must be determined definitely, what these entities are, how they are structured and which kind of symbolism is used for their presentation. 2.1 Informational Operands Informational operands are simple and complex entities in the most general sense. They have active and passive informational properties, when we say that they inform and are informed. In this manner, active components of entities can be explicated by two usual forms: as informational operands and as included informing entities within entities themselves. The included entities perform again as regular informational entities. 2.1.1 An Introduction to Informational Operands as Informing Entities Informational philosophy says that, irrespective of their physical, mental, social, individual nature, entities inform and are informed. This statement has the meaning of the fact that entities impact entities and themselves, and are impacted by entities and themselves in an phenomenal way, that is, according to the entities' proprietary possibilities of entities-concerning phenomena (e.g., physicalism, biologicism, mentalism, linguisticism, or any specific phenomenalism). On the abstract or any informational level, phenomena concerning things can be marked (specifically encoded) and structured into formulas and formula systems. Usually, an entity is informatio-nally represented by an adequate formula system in which phenomenal components of the entity occur as entities, that is informational operands, constituting together with informational operators and parenthesis pairs the so-called formula system. Markers, formulas, and formula systems are operands in the sense of informational variables if compared to adequately constructed mathematical entities. An informational operand, representing (modeling, phenomenalizing) a real entity, is informatio-nally structured, irrespective of the instantaneous 136 Informatica 19 (1995) 133-158 A.P. Zeleznikar possibilities of revealment or hiding of its informational nature. The operand structure can come to the surface through a stepwise, informationally consistent, and consequent decomposition, which is nothing else than a process of deconstruction [3] in the sense of semantic and pragmatic analysis and synthesis of the entity-structural case. This procedure of decomposition is carried of by principles (axioms, inference rules) of informational decomposition which is a part of the so-called co-unterinformational phenomenon of the entity itself and it impacting environment. Operands are representatives of simple, composed and the most complex entities of the world. In this respect, operands are operand markers, informationally well-formed formulas, and formula systems. They represent informationally arising entities in the sense of informational spontaneity and circularity, according to the principles of information [9] called informational principles. 2.1.2 Operands Marking Informational Entities Simple informational entities, marked as simple operands, are the beginning of something or something which is not informationally decomposed yet. On the other hand, arbitrary complex formulas and systems can be represented by simple markers. Definition 1 [Operands Representing Informational Entities] Informational entities are distinguished markers for simple operands, informational formulas and informational formula systems represented as operands. The so-called informational entities which hide the entity's informing are marked by small letters of the Greek or Fraktur alphabet, which can be subscribed and superscribed and written in a functional form. E.g., ; /^belief) Icti ' ' " _1 ¿.belief -a ,J. u , u t > " * ' ) ^ j «(/?)>7consdousness(C), ' '^M«) are examples of single simple, subscribed, superscribed and functional operands, respectively. □ 2.1.3 Informational Operands Marking the Informing of Entities Informing of an entity is meant to be an active component of (within) the entity, characterizing the entity's informational properties which can be observed by another entity or the entity itself. Informing of an entity is expressive (informational externalism) and impressive (informational inter-nalism). Or said by other words: informing of an entity is distinguished in two characteristic ways that belong to the basic verbal forms which are to inform and to be informed. There is no conceptual difference between informational operands as entities and informational operands as informings of entities. They are merely marked differently to distinguish them clearly between each other. Definition 2 [Operands Representing Informing Entities of Informational Entities] Informing entities of informational entities are distinguished markers for simple informing operands, informational formulas and informational formula systems represented as operands belonging to the informings of entities. The so-called informings of informational entities can hide other informational entities and their informings and are marked by capital calligraphic or Fraktur letters, which can be subscribed and superscribed and written in a functional form. E.g., A, B, • • ■, Z] 21, <£,■■•, 3; ^believe? Ca, 2l\ ©believe, 3r; A(/3), ®a(0>Cbe_conscious(£), • • • ,3z(a) are examples of single simple, subscribed, superscribed and functional operands of informing of entities, respectively. □ Operands of informing explicate the operational properties (like informational operators) to them belonging or they including informational entities. 2.1.4 Functional Informational Operands Functional operands express the informational functionalism which is an extreme generalization of function. The notion of mathematical function on the other (lower) side is a simple, reductionists notion. In the last consequence, it represents PRINCIPLES OF A FORMAL AXIOMATIC Informática 19 (1995) 133-158 137 an algorithm (mathematical definition) by which an argument set is mapped onto or into a value set, where arguments and values can be any abstractly determined objects. On the other side, informational functions are arbitrary formulas or formula systems being informationally dependent in a complex manner, e.g. as defined recursively and mutually-informingly in [13]. Definition 3 [Operands Representing Informational Functions] There are two equivalent forms of functional notation,
) C0f v In this formula, operator ^def means means by definition, Kf means is a function of or depends informationally on, etc. Operators of informational inclusion C and C0f are determined recursively, in this particular case, by /V 1= O l=of a); \ (\ V(0 Kf «) |=
^ai T(a), AserialO), ^^"^(a), • ■ ■ , ilj; rserial(ai,«2, " ' ',««), ' " " , are examples of single simple, subscribed, superscribed and functional special operands, respectively [13]. □ Definition 5 [Operands Representing Informational Inference Rules, Premises, Conclusions, and Other Entities] Many other operand symbols can be introduced marking special informational entities or to them belonging formulas. For instance, for inferential rules, their premises and conclusions, the various alphabets of small and capital letters, e.g. A, B, C, • • •, Z, a, b, c, • • •, z, A,B,C,-.-,Z can be introduced. Thus, TCP rule, i / A r> \ ^premiseC^*' ,i(A) --- inferential^ C® ÍB") conclusion^ 1 ' Operators as informational entities will be defined in the next subsection. □ where Aand B^ are variables of the premise and the conclusion function, respectively. □ 138 Informatica 19 (1995) 133-158 A.P. Zeleznikar Other specific operand symbols can be used to make entities clearly (characteristically) distinguished from each other. 2.2 Informational Operators Informational operators inform the properties of the entities to which they belong. In this sense, they are dualistic entities in regard to the informing of entities [e.g. marked by T(a) or Ia\. Like informing of an entity, the corresponding informational operator expresses the entity's property in an active informational manner. This correspondence is twofold: the informingness of the entity (informational externalism) and the informedness of the entity (informational internalism). In principle, various sorts of operators belong to an entity's externalistic and internalistic informing. 2.2.1 An Introduction to the Notion of Informational Operator An informational operator expresses the general property of an entity in the form of entity's informing. It does not mean that by an operator the entire operational possibility of an operand is exhausted. A general case operator can be particularized and universalized in many ways, depending on the happening of an operand as informer and observer. We introduce the most general operator and its possible particularizations and universalizations by the following definitions. Definition 6 [General Informational Operator as a Unique Operational Joker] The general informational operator, marked by |=, expresses the most general property of an entity, represented by an informational operand in a simple (marker) or a complex form (formula system). Although this operator is from-the-left-to-the-right-oriented, to enable its reading in the form inform(s) and are (is) informed, it does not mean that operator \= does not possesses, within its generality, the potentiality of being from-the-right-to-the-left-oriented. Thus, any particularized, universalized or direction-concerning informational operator has its ground in \= and represents nothing less and nothing more than a special case of this operator. Operator \= performs as an informational joker which can act as a substitute and (mutual) replacement of any operational phenomenon. □ A general informational operator embraces everything which can be imagined as operator, as an informational activity (informing and informed property) of the operand to which it belongs, to which it is attributed. Introducing the operational joker has the notional roots in the potentiality for leaving open any possibility of informing and determining the joker just in case when the property of an operand (informational entity) becomes (arises, emerges as) clearly identified. Thus, general informing means informing in a free and unforeseeable way, to guarantee the phenomenalism of informational spontaneity and circularity of the entity. • The operational joker implicitly expresses just this informational situation and attitude of an entity which informs and is being informed. 2.2.2 General Informational Operator and Its Particularization and Universalization Everything which is not a general informational operator, that is, |=, can be understood to be particularized or universalized through a meaning attributed to the operator. Operator particularization and universalization concerns a semantical content belonging to the operator as a consequence of the operand to which the operator is bound in an informing (externalistic) or informed (internalistic) manner. The difference between particularization and universalization is merely semantic. In fact, both mean a specialization or concretization of the operational joker. On the other hand, a particularized operator can be meaningly universalized (replaced) to some extent and up to the joker itself. Definition 7 [generally particularized and/or Universalized Operational Joker] One can introduce, together with the operational joker, four groups of four operators in the following way: — Symbols [=, . =|, are operators of informing. non-informing, alternative informing, and alternative non-informing, respectively. The alternative operators are read (from the left to the right) as is (are) informed and is (are) not informed. PRINCIPLES OF A FORMAL AXIOMATIC Informática 19 (1995) 133-158 139 — Symbols ||=, =||, represent operators of parallel, non-parallel, alternative parallel, and alternative non-parallel informing, respectively. They are read in the following sense: inform(s), do(es) not inform, alternatively inform(s), alternatively do(es) not inform in parallel. — Symbols |— , |/ , —|, -f\ represent operators of circular (cyclical, loop-like) informing, non-informing, alternative informing, and alternative non-informing, respectively. They are read in the following way: inform(s), do(es) not inform, alternatively inform(s), alternatively do (es) not inform circularly. — Symbols ||— , ||/ , —II, /|| represent operators of parallel-circular (parallel-cyclical, parallel-loop-like) informing, non-informing, alternative informing, and alternative non-informing, respectively. They are read in the following way: inform(s), do(es) not inform, alternatively inform(s), alternatively do (es) not inform parallel-circularly. □ Although, according to informational principles [9], informational entities inform in a circular way, the circular non-informing represents a particular (abstract) situation where circularity is excluded (e.g., mathematical formulas). Definition 8 [Semantically Particularized and/or Universalized Operational Joker] Arbitrary informational subscripts and superscripts for informational operators can be used. The following examples demonstrate such possibilities: L L_ alternatively 1/ alternatively railrora > XT a ) |_ in_parallel 1/ in_parallel I in-parallel la » Ifa ! F alternatively, a ' hi_circularly circularly, F in_parallel,c*' I__I_inferentially |— inferentiallyj l= by _modus_ponens etc. □ Particularization and universalization of informational operators can be chosen pragmatically, according to a language convention. 2.2.3 Informational Semicolon as an Operator of Parallelism Parallelism of phenomena belong to the most general happening of the informational. Informational semicolon, marked by ';', has the meaning of parallelism of formulas between which it is set. It can be interpreted operator-rigorously by the use of parenthesis pairs. But usually, the parenthesis pairs are omitted. Definition 9 [Semicolon Representing Informational Parallelism] A semicolon between two markers, formulas or formula systems a and (5 means that these operands inform in parallel irrespective of their mutual informational connection. In traditional logic, parallelism means a conjunction of logical operands, e.g. ahfi or a A j3. Informationally, operator || could be used. The precise definition is (a; ¡3) ?^def (a |=jn_parallel P) Thus, operator ';' can represent any of operators in-.parallel; A, ||= f=|| for an alternatively parallel case) and ||. □ Circularly parallel operators describe circularly perplexed parallel cases. 2.2.4 Operator of Informational Implication Informational implication differs essentially from the logical implication. As an operator, it appropriates the most general linguistic meaning of the verb to imply. For instance [15, 16], — to enfold, enwrap, entangle, involve; — to involve or comprise as a necessary logical consequence; — to involve the truth or existence of (something not expressly asserted or maintained); — to involve as a necessary circumstance: informer entity implies an informed entity (informational observer); — to indicate or suggest as something naturally to be inferred, without express statement; — to involve by signification or import; 140 Informatica 19 (1995) 133-158 A.P. Zeleznikar — to signify, import, mean; — to signify as muck as, to be equivalent to, to mean or intend for; — to express indirectly, to insinuate, hint at; — to assume, include (synonymously); and — to ascribe, attribute are cases of a semantic correspondence. On the other hand, to imply informationally can simply appropriate the meanings as — to interweave, interwine, interlace informationally; and — to embrace, involve informationally. Definition 10 [Operator of Informational Implication] Operator of informational implication, marked by =>, is a particularized form of the general informational operator \=, e.g. |=—>.. I^implicativelyj ^involvinglyj etc. The informationally obvious reading of operator => is 'implies informationally' or 'informs implicatively' (from the left to the right side of formula). □ Similarly as in the traditional logic, informational implication is one of the keystones of the informational reasoning and inference, by which various informational derivations can come into existence. 2.2.5 Informational Operator of Inference Informational operator of detachment in an inference rule has usually the form of a fraction line and a specific meaning. Definition 11 [Operator of Informational Inference] In an expression (informational formula) of the form where formula a is called the premise and formula (3 the conclusion, the fraction line (777) is an operator of informational inference which reads 'inform(s) inferentially' (or detachably). Thus, Ci ^inferentially /3) Premise a is marked by P and is a function of at least two operands, e.g., A and B, that is, P(A, B). Conclusion /5 is marked by C and is a function of B, that is, C(B). Thus, instead of the inferential rule R in the form R(A, B) F (A, B) C(B) there is inferentially C(B)) □ There are many "standard" forms of inference rules with characteristic premise and conclusion entities; they will be informationally examined in Section 6. 2.2.6 Operator of Informational Being-in—Informational Inclusivism Informational inclusion is a recursive concept which brings to the surface the informational connectedness or interweavement of informational entities. Definition 12 [Operator of Informational Inclusion] Operator of informational inclusion or informational Being-in is marked by C. In the context of operands a and ¡3, it is defined by the following recursive (circular) informational formula [12]: (P\=œ, \ (a C /3) ^Def a |= /3; \E(acP)J where for the extensional part E(a C /3) of the in-cludedness a C ¡3, there is, ( 2(a C /3) e V (/3 |= <*) C/3,] O M) c /3, (/3 |= a) C a, (a\=(3)ca) y The most complex element of power set V is denoted by r^/3 ,a C /3) (/3 ^ a) C /3, a; [a\=/3)c(3,a Cases, where S(a c /3) ^ 0, and 0 denotes an empty entity (informational nothing), are exceptional (reductionistic). □ PRINCIPLES OF A FORMAL AXIOMATIC Informática 19 (1995) 133-158 141 2.2.7 Operator of Informational Being-of—Informational Functionalism Informational function, as defined by Definition 3 (the informational Being-of), is a recursive concept of informational dependence between informational entities and represents a generalization of the concept of a function known in mathematics. Within a functional notation, e.g.
I =a„ must be properly parenthesized, e.g.
(•••(((=«i° K2)° K3) 0 ' ' ' l=«n-l)° l=a„,
Na1°(Nc«20(N«3 • • • 0 O«n-l0 KJ ■ • •))
To decompose an informational operator |=, for instance in a formula a \= ¡3, means to put between operands a and (3 a part of formula, that is an informational frame ([13], Definition 12), where the original formula a \= (3 becomes <&a ^ /3 3) or, formally,
(,a \= ¡3) |= by .decomposition (<£0!^/3§>)
There exist infinitely many possibilities of an operator |= decomposition. □
2.2.9 Other Informational Operators
Other informational operators can be introduced pragmatically considering a language convention and the appropriateness of operator symbols. In this sense, direct (clearly symbolical), particularized and universalized operators can be introduced. For example, |=, £, —in ?*£, C, ==>, etc. belong to the class of direct informational operators. Particularized operators, e.g. 1= inferentiallyj express a special, narrower properties of informing entities, while universalized operators express broader properties of entities which inform and are informed. There is a hierarchy of operators in the sense of the particular towards the general, which is particular-universal-general.
3 Concept of Informational Formula
Informational formula is a well-formed sequence of informational operands, operators and parenthesis pairs. The well-formedness of informational formulas is determined recursively. A formula represents an informationally arising informational entity and behaves by itself as an arising informational entity. From the philosophical point of view, a formula is nothing else than a result of an observer's informational process, which represents the observed entity at the site and through the view of the observer.
142 Informatica 19 (1995) 133-158
A.P. Zeleznikar
Informational formula is a model of the informing entity which is being analyzed, deconstructed and decomposed. This process of the informational identification of an entity through an informational formula or formula system can be continued to new forms, facts, constructions, designs, etc. according to the abilities of the observing entity. The observing entity can observe itself and perform informational changes on itself, so, this principle leads to the circular informing of an entity, that is, to the circular structure of an entity representing formula.
3.1 A General Syntax of
Informational Formulas (Operands and Operators)
Informational formula is a general term including also the system of informational formulas. A well-formed informational formula acts as an informational operand. E.g. informational markers are formulas which mark complex, composed formulas.
Definition 15 [Informational Formula Syntax] Let a mark different informational operands a, /3, ■ ■ u, A, B, • ■ Z, 21, • • 3; r, A, • • £1, A, B, • • Z; • • • and let |= be the most general informational operator which can represent any operational particularization and univer-salization. Then, an informational formula (IF for short) is informationally well-formed, if it is constructed by the following syntactic rules, where operator has the meaning 'becomes (gets, is replaced by)':
1. Operand a (as a marker) is IF.
2. Rule a <— (a) says that operand a, representing a marker, formula or formula system, can be put into parentheses. Expression (a) is IF.
3. Rule a (ai, «2, • • •, an) permits the replacement of a by a list (of mutually non-informing) operands ot\, ct2, ■••, an. Such list of operands is IF.
4- Rule a
says that a can be paralle-
lized by a where a's can represent different
entities informing in parallel. The parenthe-
(a\ .
ses can be omitted. Expression I ' 1 is IF.
5. Rule a (a |=) allows the replacement of a by a \=. Expression a \= is IF.
6. Rule a (|= a) enables the replacement of a by |= a. Expression |= a is IF.
7. Rule a (a |= a) says that operand a can be replaced by a \= a (where the parentheses are omitted), and the first and the second operand a can differ arbitrarily. Expression a |= a is IF.
This list of syntactic rules can be broadened if necessary. □
Other syntactic structures are already deduced by the defined list of syntactic rules. For instance, function y>(£) is nothing else than an expression where * is informational operator, that is,
f(0 ^ (V 1= functionally .on £)•
Definition 16 [Informational Operator Compositions] Compositions of informational operators (10 for short), where o marks the operator composition, underlie the following operator syntactic rules:
1. Symbol f= represents the general 10.
2. Operator rule |= d=ao 1=^) says that operator |= can be replaced by a meaningly adequate composition ¡=a o \=p of operators |=Q. and 1=0. An operator composition is 10.
3. If in an operator composition there are more than two operators, they must be adequately parenthesized, e.g.
ha°(l=/30 1=7) or K?)° 1=7
etc., where complex compositions are 10's. 4- Operator rule
particularly V |— universally V |— directly)
where V means 'is alternative to', says that 10 |= can be replaced by operators
h1 particularly Or |= universally "T j— directly which
are 10's. Operator (= directly represents the so-called directly expressed operators, e.g. Q, *, etc.
PRINCIPLES OF A FORMAL AXIOMATIC
Informática 19 (1995) 133-158 143
Operator compositions follow a conceptual semantics of the designing and designed entity and their syntax (parenthesizing) is determined by Definition 14- 1=1
The presented informational syntax is in no way a final case. The syntactic concepts can be refined or detailed according to informational circumstances.
3.2 Equivalence of Informational Formulas
There does not exist an informaional equivalence of different informational formulas. But, equivalence relations between different formalized expressions can be introduced on an abstract and reductionistic level, e.g. in mathematics. Within GIT, it is possible to observe different formulas with similar informational meaning (semantics, pragmatics). Thus, for example, formulas a and a \= a are not equivalent because the second formula is a derivation of the first formula in the sense of a consecutive application of modus po-nens. The meaning of a is a marker, the meaning of a |= a points to an inner circular (metaphysical, deconstructive, decompositional) structure of entity represented by a.
3.3 Implicitness and Explicitness of Informational Formulas
In concern to the discussion in the previous subsection, formula a as a marker is entirely implicit as long as its meaning is determined on some other place by some other formula. We say, that irrespective of the existence of. such other, meaningly determined formulas, formula a hides the implicitness of its informational potentiality. This means that a as any other, regular informational entity, can be decomposed into more details, determining its structure which, through decomposition, becomes more and more complex, e.g. serially as well as in a parallel manner. For example, a \= a is the first (although formally trivial) step on the way of informational decomposition. In this sense, formula a \= a informs more explicitly than does formula a. The possibility for a further explicitness of a formula does always exist.
3.4 Formula Parallelism
The possibilities of parallelism of informational formulas do always exist. The syntactic rule
is simultaneously a regular principle of an entity parallel decomposition. By this rule, parallel formulas, concerning entity a can be generated ad infinitum.
Informational parallelism is straightforward and cyclic, depending on the structure of parallel formulas.
3.5 Formula Serialism
The possibilities of serialism of informational formulas do always exist. The syntactic rule
a (a |= a)
assures the arising of serial formulas, which can be straightforward, circular or metaphysical [10]. A straightforward serial formula is, for example,
(--■(« |= /3) |= - • -V) 1= ^
and all other formulas obtained by the well-formed displacement of the parenthesis pairs. A cyclical serial formula is, for instance,
and all other formulas obtained by the well-formed displacement of the parenthesis pairs. A metaphysical serial formula is, for example,
(((((a \= x) |= <£a) |= C„) |= (S„) |= ta) \= a
where is informing, (£a is counter informing, ca is counterinformational entity, <£a is informational embedding, and ea is embedding informational entity of informational entity a. All other metaphysical formulas concerning a can be obtained by the well-formed displacement of the parenthesis pairs.
3.6 Parallel and Serial Circularity of Informational Formulas
The circularity of formulas can become very complex, for example, parallel-serial and serial-parallel.
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A parallel-serial circularity is given by a set of parallel formulas which are structured in such a manner that a certain transitivity of occurring operands through these parallel formula takes place. A trivial example of a parallel-serial scheme would be a formula system, marked by that is,
a
(a |= «i; ai |= a2;
\
an-1
o!„
\an a
A serial-parallel circularity is obtained if in a serial formula parallel subformulas appear, for instance, in the form
(•■•([aM)l=«!!)l=---a|[)l=a
where a!J, aj], ■ • • , all are parallel arrays of formulas.
The reader can imagine how this basic example can become more and more complicated.
3.7 The Case of Formula a \= /3
The case of formula a |= (3 offers a unique opportunity for clarification of the problem existing between a as informer and /3 as observer of a
Definition 17 [The Informer and Observer Problem Concerning Formula a |= ¡3] Considering the concept of operator composition in Definition 14, one has the following definition:
(a |= /3) O \=P /3)
Operator composition j=w o \=p performs as an informational transition filter between entities represented by operands a and ¡3. □
This definition explains how observer (3 can be informed about a only to the extent within which a informs in an a's specific way and ¡3 is capable to be informed in a /?'s specific way.
It seems senseful to explain the nature of infor-mingness a |= and informedness |= ¡3 additionally. The first case belongs to informational externa-lism and means that entity represented by an operator marked by a informs strictly within the informing abilities of a, that is, «-characteristically, or
The occurring operator composition f=ao |= demonstrates that the informing of a happens openly to the entire informational domain (field, space, also realm) through operator |= at the right end of the operator composition.
In the second case we have to do with informational internalism, which means that entity represented by operator marked by /3 is informed (in fact, can be informed) strictly in the framework of the informing abilities (informedness) of f3, that is, /^-characteristically, or
The occurring operator composition |= o demonstrates that the informedness of (3 happens openly to the entire informational domain through operator |= at the left beginning of the operator composition.
Within formula a j= (3 the described informational openness of the left and the right operand is blurred (however, implicitly present). Thus, possible complete meanings of the formula would be
1=) 0 (1= 0 \=p)P or a((l=a° l=)° N)° \=P P or «h«0(1= o(|= oh,))/3
where in the basic form a |= ¡3 the characteristic operator parts \=a and are implicit (invisible) and in the compositional form a\=ao \—¡3 the general operator (joker) |= is superfluous.
3.8 Inferential Informational Formulas
An inferential informational formula or inference in short has a general form
or
€
where a marks the premise (marked also by P) and /3 the conclusion (marked also by C). Thus, by ^ (or P/C), there is an inferring (informing in an inferential manner) from a to /3 (or P to C). What stands under the inferential line (informational operator of inference) is always a conclusion (operand marker, formula or formula system) and above of it, a premise (operand marker, formula or formula system). Premise means assumption,
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postulate, hypothesis, axiom, principle, and the like. E.g., a postulate is something (informational formula) taken as self-evident or assumed without proof as a basis for reasoning. Thus, a postulate within a premise performs as an axiom or as an already derived operand, theorem, formula, formula system, etc.
E.g., one cannot say that a infers /3; but, one can always observe that a informs inferentially /3, for instance in the sense, that the occurrence of a calls for an inference to /3 or that from a, there can be inferred to ¡3, etc. In this manner, the informing of something expresses the capability or characteristics of the informing entity in respect to the informed entity.
In case ^ we have the situation which must not be forgotten:
(a I—q:, inferentially 0 |—/3, inferentially/3)
dicates (logical functions concerning elements as functional arguments belonging to arbitrary sets) in the value domain of truth and falseness (untruth). Informational entities and informational functions concern formulas which, within them and in parallel, can produce formulas, as results or "values". Let us demonstrate the difference between both approaches on the level of existence of something, truth of predicates, and informing of something.
Something, marked by a, certainly has the property of existence. The framed expression something exists is a formula which transits in
a predicate form
something exists
is true
a formalized way, the predicate form
In
a exists is true
If a |=«, inferentially, there can be not only \=0,inferentially P, but any other kind of conclusion, say, |=7, inferentially 7: another, informa-tionally different (logical) structure of 7 in comparison to /3. Rules of inference can arise as any other regular informational formula. If one proceeds from standard inferential rules (e.g., tertium non datur, modus ponens, modus tollens, etc.), it does not mean that arbitrary inferential rules cannot come to the theoretical surface or cannot emerge during the theoretical discourse.
Further, it must be clarified what can the application of an inferential rule p (or E) upon an in-formationally approved formulas a and /3 (acting as a premise P and conclusion C) mean, for instance, in the form of the informational Being-of or functionalism [or R(P, C)]. In this case,
inferential rule p becomes an informational function over formulas (formula systems) marked by a and /3.
4 The Propositional and the Predicate versus the Informational
4.1 Traditional and Informational Logic
Theory of logical propositions and predicates (for instance, [5, 8]) introduces propositions and pre-
corresponds to the predicate E(a). A predicate is always understood to be a matter of the observer, e.g. mathematician. On the other hand, informational formula which expresses the fact 'a exists' or, more precisely, 'a informs to exist', or 'a informs existingly', that is, a |=exist5 belongs not to the observer, but to the informer a as a property of its informing. The predicate form E(a), expressed in an informational form, would be
(a h= exist) 1=
true
The truth of a predicate concerns the predicate and not the entity as an argument of the predicate. Thus, the framed expression
entity exists
informs true
can be understood as a predicate E(a) with an implicit (assumed) faculty of trueness on one side and as an informational formula (a |=exist) |=true which transparently (expressively) informs the faculty of the entity itself, for example, in the form as: entity is, as it does exist, and as it does exist in a true way.
The other (contrary) case is
entity does not exist
informs true
which corresponds to the predicate form E{a) and to the informational form
(a exist) |=
true
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In this point, one has to clarify how does entity a (in German, das Seiende a) not exist. The answer is: in a certain way! Informational operator b^exist is in respect to operator |=exist nothing else than a particularized operator of the type (=exist-This is uniquely not clear in case of E(a).
The difference which can now be drawn between the predicate view and the informational one is the following: the predicate view concerns implicitly the observer of an entity while, on contrary, the informational view concerns explicitly the informer and the observer and where both can perform in one and/or another way, that is, informer as informer and observer, and observer as informer and observer, simultaneously. In the informational case, the observer must be explicitly present (marked) and must inform and be informed explicitly, through concrete, particularized or universalized informational operators. While the predicate case concentrates on the observer and the informer (the argument or variable of the predicate) is only an object of the observer, the informational view distributes the informing between both the observer (the informedness) and informer (the informingness).
Axiomatically, the informer stands before the observer and the observed (the informing entity, that is, informer) can only be that which informs. Both are informationally active and passive entities (subject and object, simultaneously) and explicitly present (informationally determined).
Propositional and predicate logic stress the observer's view, that is, the so-called informational internalism. Informational logic unites the so-called informational externalism and internalism in the framework of informational metaphysica-lism and phenomenalism. And, in this kind of view lies the novelty and the power of informational arising as a spontaneous and circular phenomenon, within the discourse which is on the way and which follows.
4.2 The "Value" of an Informational Formula
The concept of value belongs to the basic mathematical concepts. In mathematics [15, 16], value is the precise number or amount represented by figure, quantity, etc. For instance, numbers, elements of sets, truth and falsity, magnitude, a point in the range of a function, the value of a word,
etc. are values for variables and functions. A similar question can be reasonably put to the surface in case of the informational: which are the values of informational operands as variables, markers, formulas and formula systems? How can informational values be achieved (accessed) and what do they represent as informational formulas?
In music, value is the relative length or duration of a tone signified by a note. In painting, it is due or proper effect or importance; relative tone of colour in each distinct section of a picture; a patch characterized by a particular tone. In philosophy, value means axiology.
An informing formula produces formula-like results. A result can be understood as a part, piece of formula, as an arising parallel formula or formula system, which value is a semantically and pragmatically converted, transverse sort of information, e.g. text, picture, voice, signal, etc. The same principle can be used for the domain of input operands, that is, informational variables, formulas, etc. as informing entities.
Informational formulas are, in respect to the natural means (languages, pictures, voices, signals, etc.), adequately informationally encoded entities which, in any state or position, can be under standingly decoded as values, in a "natural" form. Informational encoding and decoding can use any formalized (mathematized, systemized, procedural, etc.) means, methods, concepts, algorithms, apparatuses, approaches, formulas, etc. as well as those of the informational view, science, theory, systems, etc.
4.3 Logical and Informational Examples
Tautology and (informational) circularity are the focal problems of traditional mathematical and informational logic. We will show how a syntactically circular formula in mathematical logic is never comprehended as a circular (tautological) scheme while within an informational logic just this type of syntactic expression is considered to be circular. Thus, traditionally, the formula circularity (tautology) is pushed off the conscious horizon, while informationally circularity is considered as a cyclically operating kind of informing. To get a clear picture of such phenomenalism, we will use examples concerning the so-called foundations of mathematics, that is, its metatheory.
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Example 1 [Implicative Axioms for Pro-positional Calculus] Formalizing the logical reasoning (inference) in propositional calculus ([5] p. 66), Hilbert lists a (geometrical) group of his axiom formulas of implication:
(■B - A), (A -> B),
((i? —» C) —» (A O)
(A ->(A->B)).
(A^B)
Informationally, these implicative formulas exert a sort of circularity regarding propositional operands A, B and C. □
The first formula says that proposition A, if not an axiom, has its logical cause in a proposition B. It simply means the following: Something implies that it is implied by something other. The second formula stresses that if proposition A implies an implication A —» B, then A implies B. It can be interpreted as: If something implies that it implies something other then something implies something other. The third formula says: if A implies B then the implication B C implies also the implication A —► C. Said by other words, there is: Something implies something other implies the following: if something other implies something third then something implies something third.
The listed axioms are in a certain accord with common sense. All of them are identically true logical formulas, which can be easily verified by
V BV A,
V ÂV B,
V (B AC) VÂVC
The first formula is circular in a a, the second one in ola and (3b, and the third one in aa, (3b and 7c-All together form a parallel informational system (operator ';').
Example 2 [Implicative Axioms for Informing of Informational Entities (Operands)] An instructive, now informational case with informational implication operator
is
(a
(a (a
a ß)) ß)
(a =
iß (a:
a);
ß); ' 7) : 7))
which leads to the basic informational axioms, for example, of the form
(a;
(a: (a
(« b) = =» (a^(ûb);
(«b))) = (a =>■ (a b));
(«b)) =
0=«3>=>
((a (b «)))
_A (A A B) (A A B)
respectively.
Reading the original formulas, a mathematician does not observe the circular structure of the listed axiomatic formulas. Implication seems to be such a kind of the logical operator which does not evoke the 'feeling' of circularity although the markers of one and the same kind are used several times in the implicative expression (e.g. in implication of implication). This fact becomes informationally true if in original formulas the logic implication operator is replaced (universalized) by the informational joker and operands are adequately marked by a a, (3 b and 7c that is,
a A f= (PB |= otA)\
(aA \= (aA b Pb)) b 0,4 1= (3b);
(<*a b M h (((3b b 7c) h («A |= 7c))
The last example shows how the "global" Hil-bert's implicative axioms can reasonably be applied in the informational case where the traditional Truth of Propositions is replaced by the Informing of Informational Formulas. □
5 Phenomenalistic Axioms of the Informational
Zur Erleichterung soll bei den ersten Axiomen die sprachliche Fassung hinzugefügt werden.
—D. Hilbert und P. Bernays [5] 5
The so-called phenomenalistic axioms of the informational are meant to be the axioms of the object and metatheory, and the inference axioms (initial rules for informational inference) underlie the general informational phenomenalism. General informational theory is, namely, a unit of the formal object theory and the formal metatheory, that is, the theory of informational inferring (proving, causing, concluding—deriving). As stated in the quotation of this section, the natural language comprehension cannot be avoided at the very beginning of the presented informational axioma-tization.
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In this section the basic axioms will be presented in an aprioristic and postprioristic manner. The independence of axioms will not be considered. Later on, it will become clear that only one informational axiom can be chosen, however, by the use of informational inference rules, other axioms can be derived (deduced).
5.1 Informational Externalism
Let us try to state which sort of axiom could be quite on the top of the informational. Already in Example 2 we have applied Hilbert's axioms [5] for the informational case.
Axiom 1 [Informational Externalism] A-prioristically (commonsensically, trivially, intuitively [6]) at the top of the informational (system) has to be something which deepeningly (most essentially) concerns an informational entity a. So, let it be an informational implication of the form
(a |=))) (a => (a |=))
This axiomatic formula says: "If informational entity (represented by operand a) implies that it implies its informing(ness), then the entity implies that it informs." The next, substantial axiomatic rule of informational externalism (according to Example 2 [5]) is
(a |=) (« (a |=))
Informing (ness) of a implies that a itself is the cause of its informing (ness). □
According to the last axiomatic formulas, everything informational, irrespective of its informational structure or complexity, informs. Both formulas are informationally (and traditional-logically) consistent, that is, informationally (logically) noncontradictory. In traditional logic, pertaining to truth, it would mean, that both formulas are true (even identically true) or, expressed informationally, (a (a (=)) ^true-
Example 3 [Externalism of Non-informing] Informing and non-informing of an entity a are parallel phenomena. Non-informing may be comprehended as a particular phenomenon of informing. Thus, operator which reads 'does not informis a particular case of operator \=.
a |= means that a informs in a specific manner, that is a-characteristically. a means that a does not inform in a certain way, that is, it informs a-non-characteristically. Thus,
(a (a 1=0«) and (a ^ (a \£a)
Operator would mean informs differently in comparison to a-characteristically.
According to Axiom 1, for a particular case concerning a, there is a =>• (a \= particularly)- According to this principle, also
a => (a
holds. This implication will become significant in our further discussion. □
5.2 Informational Internalism
Informational internalism is a dualistic concept in regard to informational externalism. Axioma-tically, the question arises, which of both phenomena is the primary one and which is the consequence of the other. Thus, quite at the beginning of axiomatization, the next axiom could also be accepted.
Axiom 2 [Informational Internalism] A-prioristically (commonsensically, trivially) at the top of the informational could also be something which deepeningly (most essentially) concerns an informational entity a in the sense of its infor-medness. So, let introduce an informational implication of the form (a (a (|= «))) (a (^ a)) This axiomatic formula says: "If informational entity (represented by operand a) implies that it implies its informedness, then the entity implies that it is informed." The next, substantial axiomatic rule of internalism is, for instance,
(1= «)=►(«=* (N «))
Informedness of a implies that a itself is the cause of its informedness. □
According to the last axiomatic formulas, everything informational, irrespective of its informational structure or complexity, is informed. Both formulas are informationally (and traditional-logically) consistent, that is, informationally (logically) noncontradictory. In traditional logic, pertaining to truth, it would mean,
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that both formulas inform true (are identically true), or expressed informationally, (a =>• ()=
a)) btrue-
5.3 Informational Metaphysicalism
Informational metaphysicalism is a general and entity specific way of circular informing. In general, it proceeds from the initial circular form a b o. which is trivially circular, but becomes structurally circular by decomposition. Specifically, the decomposition of this form can be standardized to some extent, introducing explicitly the components of informing, counterinforming and informational embedding as entities which inform within an informational entity [10].
Axiom 3 [Informational Metaphysicalism] Aprioristically (commonsensically, trivially, intuitively) at the top of the informational could also be something which deepeningly (most essentially) concerns an informational entity a in itself, as its inner informing or informational arising, called metaphysicalism. So, we can introduce an informational implication of the form
(a =>■ (a (a b a)D (a ^^ (a b a))
This axiomatic formula says: "If informational entity (represented by operand a) implies that it implies its metaphysicalism, then the entity implies that it informs and is informed circularly." The next, essential axiomatic rule of metaphysicalism is, for instance,
(a b °0 (a ==> (a b a))
Circular informing within a itself implies that a itself is the cause (phenomenon) of its metaphysicalism. □
According to the last axiomatic formulas, everything informational, irrespective of its informational structure or complexity, informs and is informed in a metaphysical manner.
5.4 Informational Phenomenalism
Informational phenomenalism means a parallelism of informational externalism and internalism regarding an informational entity a. By this, an entity is open as an informer and observer to its
environment and to itself (metaphysicalism). Informational phenomenalism is the most general concept of informing of entities. This belief can lead to the axiom which follows.
Axiom 4 [Informational Phenomenalism] Aprioristically (intuitively, commonsensically, trivially) at the top of the informational could also be something ivhich deepeningly (most essentially) concerns an informational entity a toward the outside (outward, externally), toward the inside (inward) and in itself, as its entire informing or informational arising, called phenomenalism. So, we can introduce an informational implication of the form
a
ce
This axiomatic formula says: "If informational entity (represented by operand a) implies that it implies its phenomenalism, then the entity implies that it informs externalistically and is informed internalistically." The next, essential axiomatic rule of phenomenalism is, for instance,
Phenomenal informing of a implies that a itself
is the cause (phenomenon) of its phenomenalism. □
According to the last axiomatic formulas, everything informational, irrespective of its informational structure or complexity, informs and is informed in a phenomenal manner.
It will be shown how Axioms 2, 3 and 4 can be derived form Axiom 1 if the axiomatic inference rule of informational modus ponens is adopted.
6 Axioms Related to Informational Rules of Inference
Inference rules of a theory pertain to the theory's metatheory, which performs as a theory of theory. . In this function, a metatheory concerns the proving, founding, logicism and formalism of a theory, that is, in regard to metatheory, the object
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theory. Separation between the object and me-tareasoning is traditional and roots in mathematics, in its platonistic (tautological) approach with the intention to make theories function in a non-contradictory, logically consistent and reductionists way.
6.1 The True versus the Informational
Truth is the central concept of any mathematical theory and of mathematics as such. Everything derived from axioms by rules of inference must be true. Through mathematical proofs, the truth of theorems or derived consequences must be verified. Otherwise, the derived results are not mathematically correct. The basic question is how this traditional approach could be diversified in such a way that the mathematical truth becomes only a particular informational entity (operand) or an entity's property (operator)?
In Subsection 4.3 we have shown a possible difference between the true and the informational. The difference can exist in the following different manners:
N The true The informational
1 Logicism Informationalism
2 Particularization Generalization
3 Tertium non datur Various informing
4 A is true or a is informational
false
5 A informs true a informs
6 A |=true a |=
7 A informs false a does not inform
in a way
8 A |= false Or atf
A l^true
9 A is informed a is informed
true
10 l^true A [= a
11 A is not informed a is not informed
true
12 l^true A or
^false A
The last list of differences illustrates only the initial possibilities; so the reader can continue to list further imaginable differences.
6.1.1 Identical Truth of Propositions and Predicates
Propositional formulas (which are propositions representing logically connected propositions) can be constructed in such a way that they do not depend on the true and false values of their operands. Such formulas are said to be identically true or identically false. For instance, propositional formula A —> (B —> A) is identically true, while formula (its negation) A —»• (B —> A) is identically false.
The triviality of logical axioms and rules of inference lies in their identical trueness. For instance, the pure implicative axioms of logic ([5], p. 66) are identically true, that is, they do not depend on the values of their propositional arguments. The same is valid for the derivation (deduction) rules of the type modus ponens and modus tollens, which can be logically transcribed into (AA(A -+B))-*B and ((A B)AB) B, respectively. In this way, something hidden (unre-vealed, intuitive and, also, tautological) remains in the background of these rules of inference.
6.1.2 The Value Domain of the Logical
Following the principle of tertium not datur, there are only two values of propositions and predicates in traditional logic, that is—true and false. In multivalued logic, more than two values are permitted and gradations between true and false value are possible. But the nature of the principles of trueness remains preserved in various manners (e.g., probabilistically, modally [2], etc.). A valuation is an assignment of truth values (T and i.) to the proposition sentences after their semantic analysis.
6.1.3 The Formula Value Domain of the Informational
The informational formula value domain is formula-like. Arguments and values of informational formulas are formulas as inputs and outputs. The difference to the traditional-logical is that arguments can influence the entity in question to an informational extent (in trivial cases also to an 'entire' extent) and that the so-called values (results) are 'produced' (influenced) only to some informational extent (trivially, to a full extent).
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Informational formulas simply absorb the prepositional and predicate power of a traditional-logic apparatus (calculus).
'To influence to an informational extent' means to impact something informationally not as a product but as an already existing entity; and the similar concerns the informational impacte-dness, where something performs its influence on the entity impacted by it.
6.2 Mathematical Implication versus Informational Implication
As the mindful reader can observe, the mathematical implication is not only logical. In fact, the implication is a metamathematical (philosophical, intuitive) connective of arguments, closely tied to the semantics of each argument and the implication as a semantic structure in particular.
Informational implication approaches to various dictionary (informational) concepts of the word 'implication'. It certainly absorbs the concept of mathematical implication.
6.3 Rules (Axioms) of Informational Modus ponens
Modus ponens (MP for short) belongs to the most popular rules of inference as a mechanism for mathematical deduction of formulas from an object theory axioms and already deduced particular formulas, called theorems. Simultaneously, modus ponens is the main instrument in the proof procedures which are nothing else than just deduction processes as described. It considers the conveyance (an old law) and is the way in which anything deduced is obtained. MP is a mode of deductive operation (e.g. modus operandi or modus agendi).
The Latin verb pono (posui, positum) means to set down, before; to lay out, put out at interest in the sense to lay down as true, assert, assume, etc. MP is a mood that affirms (in German, bejahender Modus). It is a rule by which from if p then q together (and) with p, the operand q may be inferred. The full meaning in Latin is modus ponendo ponens or law of detachment (in German, Abtrennungsregel), written in the form (p,p —> q) —► q. The meaning is: if, simultaneously (in German, sowohl), p and also 'if p then q' is valid (true), then also q is valid (true). Because
MP is a rule, the rule arrow —> has to be used as a communication sign for an action (operation) instruction.
6.3.1 Interpretation of Logical Modus ponens
In traditional logic, modus ponens (the rule of detachment) has the form
P,P-» g 9
where the inferring line is the operator of detachment. As already shown, this rule (when neglecting its communication role) represents an identically true formula in the form (p A (p —»
The uttermost informational interpretation of the above formula regarding the truth as the only relevant logical value could be the following:
( (p [—"true5 {P q) btrue) |=true\ ■
I -p- 1 l" true
V q |—true J
In the last case, by the entering into formula of MP, the truth of q has to be verified. If q is a true theorem, then in the premise, p and p —»■ q are assumed to be valid, that is, true. Such ail understanding of MP seems to be commonsensi-cal. However, MP informs true without regard to the truth of the constituting components (p and p q).
6.3.2 Informational Modus ponens and Its Possible Interpretations
Together with the fundamental axiom of the object informational theory, that is, a (a |=), we need a fundamental inference axiom, by which from the initial informational entity a the result a |= can be derived (deduced).
Inference Axiom 1 [informational modus ponens] We adopt the following basic inference axiom for informational derivation:
g;(q /3)
P
By this rule, marked by Rmp(o;,/3)J where 'mp' in the subscript stands for 'modus ponens', operand (formula, formula system) (3 will be derived from operand a. (formula, formula system), that is,
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a mp P or, simply, a ¡3
Formula a -» (3 is called derivation (by modus po-nens) from a to /3. □
By means of the last inference axiom from the first object axiom (e.g., Axiom 1) a theorem can be proved which follows.
Theorem 1 [Externalistic Informing of an Informational Entity] If a is an informational formula, then a \= is an informationally regular (a-equivalent, a-replaceable) formula. It means that in decompositions of a (serial, parallel, circular, metaphysical or whichever de-construction), formula a (= performs as another phenomenon of formula a. There is, certainly, a -» (a \=). □
Proof 1 [Formula
a |= as a Regular Occurrence of a] The initial 'axiom' of an informational operand a is the operand itself. We must prove, that formula a |= is derivable from a. Axiom 1 offers the informational validity of formula a (a |=). In this way, we dispose with elements of the premise necessary for modus ponens. Finally,
a; (a (a [=))
a b
In fact, this is a trivial (aprioristic) proof of the informational existence of a J= if the existence of operand a was axiomatized. Thus, the existence of derivation a -»(a |=) is proved. □
Theorem 2 [Internalistic Informing of an Informational Entity] If a is an informational formula, then |= a is an informationally regular (a-equivalent, a-replaceable) formula. It means that in decompositions of a (serial, parallel, circular, metaphysical or whichever deconstruc-tion), formula b a performs as another phenomenon of formula a. There is, certainly, a-» (|= a). □
Proof 2
[Formula (= a as a Regular Occurrence OF a] We must prove, that formula \= a is derivable from a. Axiom 2 offers the informational
validity of formula a (|= a). Let us shoiv cases of proving the derivability of b a. The first possible case (Axiom 2) is
a; (a =» (|= g)) \= a
The second possible case considers the axiomatic fact (a |=) (|= a) as a necessity which says that if something informs, something must be informed. Thus,
(a
ha
At the end of the proof, let us show informationally three axiomatic implications which follow according to Example 1, the third rule:
(a (a |=))
(((a |=) (|= a)) =► (a (|= a)));
(a (|= a)) =»
((O a) (a b)) =►(«=»(<* |=)));
((a 1=) (b a)) =»
((fl=a)=*a)=>([a b) «))
The implicative circularity of entities a, a |= and |= a is complete.
Thus, the existence of derivation a -»(|— a) is proved. □
Theorem 3 [Phenomenalistic Informing of an Informational Entity] If a is an informational formula, then formula system a |=; |= a is an informationally regular (a-equivalent, a-replaceable) formula. It means that in decompositions of a (serial, parallel, circular, metaphysical or whichever deconstruction), formula system a bi |= a performs as another phenomenon of
formula a. There is, certainly, a -»(a bi b a)-□
Proof 3 [Formula System a bi b a As A gular Occurrence of a) A consequence of the previous axioms and theorems is formula
a =>■ (a b; b a) By informational modus ponens, there is, a; (a =» (a bib "))
a bib a
This proves a -» (a bi b D
PRINCIPLES OF A FORMAL AXIOMATIC
Informática 19 (1995) 133-158 153
Theorem 4 [Metaphysicalistic Informing of an Informational Entity] If a is an informational formula, then formula a |= a is an informationally regular (a-equivalent, a-replaceable) formula. It means that in decompositions of a (serial, parallel, circular, metaphysical or whichever deconstruction), formula a \= a performs as another phenomenon of formula a. There is, certainly, a -» (a |= a). □
Proof 4 [Formula a\=a as a Regular Occurrence of a] A consequence of the previous axioms and theorems is formula
(a |=; |= a) =>- (a |= a) By informational modus ponens, there is, (a |=; |= a); ((g |=; |= q) (g |= q)) a \= a
(See Definition 15 for a \= a.) This proves a (a \= a). □
Consequence 1 [Replacement Possibilities for an Informational or Informing Entity] Let us have
a, b £ {a, a (=, |= a, a |= a, (a |=; |= a} Then, for a ^ b, there is a -» b and, so, a 6. □
Consequence Proof 1 [proving a <— 6 for
Entities and Their Informing] The last consequence means the possibilities of replacements in decomposition (deconstruction) procedures, that is,
a
; (a H
(1= a)
■ (a\= a)
According to Subsubsection 6.3.1 where the logical modus ponens was informationally interpreted, we can interpret the informational modus ponens in several informational manners. The first possible and informationally consequent interpretation of MP is externalistic and yields
(<*!=;(<»=> I)
where a and /3 can be arbitrarily complex formulas. As we see, its components (three of them in the premise, one in the conclusion) and the entire rule of MP inform in an externalistic manner. The exact meaning of this interpretation could be the following: if a informs its informational existence and, in parallel, the implication a implies /3 informs its informational existence, then /3 informs its informational existence. Under these conditions, the respective formula of MP informs its informational existence.
Other interpretations of informational MP could be internalistic, metaphysicalistic and pheno-menalistic. The consequent internalistic interpretation of MP is
t=(|= a; [=(<*=>/?)) ' M
The consequent metaphysicalistic interpretation becomes pretty cumbersome, that is,
(a |= a; (a =» /3) |= (a /3)) |= (a |= a; (a => /3) |= (Q /3))
/3 1=/3
h
/a, \ a |=, a |= a,
(a, \ a |=, 1=
\a b a J
The consequence can be proved by considering the previous axioms and theorems. □
(a \=a-,(a=> ¡3)\= (a ¡3)) (= (a |= (q => /3) [= (q => /3))
/3 1=/3
where metaphysicalism must be considered on all five components of the inference rule (metaphysicalism of two components of the premise; the premise as a whole; the conclusion; and the rule as a whole).
The consequent phenomenalistic interpretation is cumbersome too, which becomes evident from the two equivalent inference rules, where the first one informs and the second one is informed:
154 Informatica 19 (1995) 133-158
A.P. Zeleznikar
f(a\= ;|=a); \
«=►/3) h !=(«=►/?)) H
(a 1=; |= a)
|=; 1= /?
h
^(a |=;|= a);
(=((a=»jS)|=;l=(«=>/3))i/
N
N
' ((a /3) N N (tt =»/î]|) H ^ !=««=>/?) h 1= («=»/?»)!/ /3 h M
The reader can imagine how mixed externalistic, internalistic, metaphysicalistic and phenomenali-stic interpretations are possible. In this way an informational explosion of MP possibilities exists.
6.4 A Generalization of Informational Inference Rule Interpretation
The case of informational MP interpretation calls for a generalization principle in the following sense.
Inference Axiom 2 [Inference Rule Phenomenalism] An inference rule
K(A,B)
P(A, B) C(B)
fR(A,B)b \
|= R(A, B), R(A,B) R(A, B),
,B)hNK(A,B )))
R(A, B)
Concerning premise F(A, B) and conclusion