Jakościowa teoria informacji (‘A Qualitative Theory of Information’; Wydawnictwa Naukowo-Techniczne, Warsaw 1970, 223 pages) asks a question that classical information theory passes over: what information itself is, and not merely how much of it there is. Marian Mazur does not start from any ready-made definition. He first takes the process of control apart, then names what has emerged from this analysis, and only at the very end checks how his concepts relate to Shannon's well-known formula.
The book has thirteen chapters, followed by classification tables in three languages, summaries of the whole in Russian and in English, and an index. The theory is set out in the order definition, theorem, proof: 141 definitions numbered by chapter and 33 theorems with proofs, almost always based solely on what has been defined earlier. On top of this come 38 figures — mainly diagrams of paths on which arrows show what passes into what — and a wealth of short, elementary examples from measurement, engineering, language and everyday life. All the definitions are in the dictionary of this site; the full list in the order of the book, with numbers and pages, is shown in the definitions in order view. Below we discuss how the theory is built and where it leads — the definitions are in the dictionary, and here we only link to them.
The structure of the book can be described in four steps. Chapters 2–6 build the apparatus: control and the path, the message, transformation, and then two applications of transformation — along the path (code) and across it (information). Chapter 7 joins them in the concept of informing, Chapters 8–12 classify informing by the structure of the coding chains and carry this classification one level higher, and Chapter 13 returns to the quantitative measure. Repetition is characteristic: the same set of variants (trivial, identity, equality, inverse, operational, principal, associative, chains and resultants) passes from transformation to code and to information, and three variants — simulative, dissimulative and confusive — run through the successive kinds of unfaithful informing; hence whole families of terms in the dictionary.
From the Publisher and Preface (pp. 9–12)
The publisher's note explains the method and the title. The publisher points out that the starting point is not existing information theory — the author reaches it only at the end of his own route; the adjective ‘qualitative’, in turn, refers not to the value of information but to its kinds, in the sense in which a chemist distinguishes qualitative from quantitative analysis. In the preface Mazur places his theory beside the quantitative theory (highly developed) and the value theory (in its infancy, linked to decisions), lists the earlier works from which it grew — among them a chapter of Cybernetyczna teoria układów samodzielnych (‘A Cybernetic Theory of Autonomous Systems’) — and sums up the central idea in a single distinction: codes run along the path, items of information across it.
Concepts introduced in the chapter: none — this part of the book contains no numbered definitions.
Chapter 1. Introduction (pp. 13–26)
The introduction justifies the whole undertaking: information, without which cybernetics cannot do, has so far not been strictly defined by anyone. Mazur does not challenge the theory of Shannon and Hartley — he praises the way in which the measure of the amount of information was established — but he points out two shortcomings in it: it does not say what information itself is, and it gives no measure where there is no set of possible events and no probabilities. Even the name is misleading: it is easy to mistake the ability to count bits for knowledge of what information is. A survey of the literature serves as evidence: some of the authors cited admit that the name of the theory misleads or that its groundwork is uncertain, while others specify information in a vicious circle, through a vague ‘content’ or through the psyche of the recipient, which cannot be carried over to machines. The method of the book is announced by a distinction between two routes: looking for a definition of a word already in use leads to fruitless disputes, whereas the scientific way is first to determine the concept needed for the inquiry and only then to choose a name for it; such a terminological convention may be clumsy, but it cannot be wrong.
Concepts introduced in the chapter: none — this part of the book contains no numbered definitions.
Chapter 2. Control (pp. 27–32)
The chapter ends with the question that guides the rest of the book: what do the intermediary paths between the controlling system and the controlled system do to the process of control? To pose it, Mazur chooses a domain that is sure to contain everything people call information — control, that is, striving towards a goal — and treats telecommunications, psychology or linguistics only as a source of examples, not of proofs. He recalls the five basic concepts of cybernetics and stipulates that the roles in a closed loop are fixed only by someone outside it who cares about the result; this need not be a being with a psyche — a system with its own source of motivation is enough.
Concepts introduced in the chapter (5): controlled system (2.1), controlling system (2.2), control path (2.3), control loop (2.4), control process (2.5).
Chapter 3. Message (pp. 33–40)
The chapter introduces the first building block, the message, and assigns it its place in the path. An important remark concerns time: the image need not arise after the original, because control can be based on the past, the present or a forecast, and the direction of the path is determined by accessibility, not by the order of events. The starting point is the observation that a physical state can be grasped from two sides: that of energy and mass (how much of something there is) or that of structure (how it is arranged); working processes concern the former, control processes the latter. Whether something is a message is decided solely by distinguishability, so a single set may contain things that are not alike at all. Messages form transverse and longitudinal sets, with originals at the beginning of the path, images at the end and intermessages in between; Mazur shows the wide reach of these concepts with examples from measurement, engineering, language and cartography. The last distinction divides messages into active ones, which last only while energy flows, and passive ones, that is, traces, reading which costs the recipient energy and carries a risk of distortion.
Concepts introduced in the chapter (8): source of action (3.1), receiver of action (3.2), message (3.3), transverse set of messages (3.4), longitudinal set of messages (3.5), original (3.6), image (3.7), intermessage (3.8).
Chapter 4. Transformation (pp. 41–59)
This is the chapter of tools and the base of everything that follows: here the apparatus is created that Mazur uses to the end of the book, together with three theorems — among them the one stating that a transformation combined with its inverse transformation changes nothing (gives a trivial transformation). The final part sets the apparatus beside logic and points out what the theory of relations lacks: the breakdown of a transformation into an operation, its kind and its parameter — needed because in control a relationship usually has to be reconstructed rather than received ready-made — and the requirement of a single result, which Mazur derives from the physical nature of the process. Using examples from logic textbooks, the author shows that whether a relation is accepted as transitive may depend on how language names the relationship in question, and that transitivity in general need not hold in particular cases; here the calculus of transformations gives a result that is certain and independent of language.
The apparatus itself is simple and deliberately his own, and it is built around an asymmetry: anyone who knows the transformation — the turning of one message of a pair into the other, always with a single result — will get from the first message to the second unambiguously, but cannot guess the transformation from the pair alone. When there are many pairs, one looks for a transformation common to all of them — the principal transformation, on which Mazur also bases the unambiguous determination of the inverse one. Any transformation can, moreover, be broken down into operations, and an operation into a kind (a qualitative feature) and a parameter (a quantitative feature). Beside transformations that indicate what has to be done stand associative ones, which say only what corresponds to what.
Concepts introduced in the chapter (23): association of messages (4.1), transformation (4.2), primary message (4.3), secondary message (4.4), non-trivial transformation (4.5), trivial transformation (4.6), identity transformation (4.7), equality transformation (4.8), inverse transformation (4.9), operation (4.10), kind of operation (4.11), parameter of operation (4.12), operational transformation (4.13), principal transformation (4.14), inverse principal transformation (4.15), inverse operational transformation (4.16), inverse operation (4.17), inverse kind of operation (4.18), inverse parameter of operation (4.19), associative transformation (4.20), inverse associative transformation (4.21), transformation chain (4.22), resultant transformation (4.23).
Chapter 5. Code (pp. 60–69)
A code is a transformation along the path, and the chapter carries over to it the whole set of variants from Chapter 4, from the trivial code to the coding chain and the resultant code. Two theorems state that successive codes combine into a single resultant code, and that a code combined with its inverse code cancels out; Mazur draws from them several practical conclusions for engineering and telecommunications.
Most space is given to a comparison of two ways of coding, decided by a reckoning of convenience: the number of entries in the table on one side, the intricacy of the rule on the other. With huge listings for which no rule has been found, the very task of locating an entry becomes the problem — in this Mazur sees the reason why machines are harnessed to such sets. Associative codes, that is, listings of the ‘this corresponds to that’ type, are convenient, but helpless when faced with entries outside the listing and worthless once reality changes; a general rule replaces a whole listing with a single prescription and is the more valuable the more cases it covers.
Concepts introduced in the chapter (14): coding association (5.1), code (5.2), non-trivial code (5.3), trivial code (5.4), identity code (5.5), equality code (5.6), inverse code (5.7), operational code (5.8), inverse operational code (5.9), principal code (5.10), inverse principal code (5.11), associative code (5.12), coding chain (5.13), resultant code (5.14).
Chapter 6. Information (pp. 70–80)
Here comes the definition of the concept to which the book is devoted: information is a transformation across the path, that is, a transformation linking messages that occur at the same place of the path. The last two parts of the chapter matter for what follows: the search for principal information in data (a short sequence fits many rules, a longer one fits a single rule or none, and measurement errors make the matter harder) and an explanation of why associative information is unambiguous only when the choice is between two messages — in this Mazur sees the origin of the bit. The everyday phrase ‘a pair of messages contains information’ is thus given a strict sense: what is meant is the transformation linking them. The chapter repeats on transverse sets the range of variants known from Chapters 4 and 5, and with examples from measurement and from language shows how far the new concept reaches. The theorems state that items of information in a chain add up, and that a change undone by the inverse change leaves no trace.
Concepts introduced in the chapter (14): informational association (6.1), information (6.2), non-trivial information (6.3), trivial information (6.4), identity information (6.5), equality information (6.6), inverse information (6.7), informational chain (6.8), resultant information (6.9), operational information (6.10), inverse operational information (6.11), principal information (6.12), inverse principal information (6.13), associative information (6.14).
Chapter 7. Informing (pp. 81–86)
The chapter is the keystone of the book: it joins code with information and lays out the plan of the next three chapters. It also returns to the distinction from Chapter 3: one and the same measurement is at once work and informing — depending on whether we look at energy or at structure; in a working process energy serves to perform work, in a control process it is solely a cost, ideally zero. Formally, code and information are the same kind of transformation; which of the two it is depends solely on its position relative to the path. Carrying over that by which the originals differ from one another onto that by which the images differ from one another is informing; its simplest case requires four messages. The central theorem 7.1 presents the information in the images as the information in the originals enclosed on both sides by codes — the inverse one and the ordinary one — so the controlling system reacts to what has reached it, not to what happened at the source; the kinds of informing discussed in Chapters 8–10 depend on the structure of the coding chains (complete and separate; complete but not separate; separate but not complete).
Concepts introduced in the chapter (4): informing (7.1), simulative informing (7.2), dissimulative informing (7.3), confusive informing (7.4).
Chapter 8. Transinforming (pp. 87–117)
The longest chapter asks when informing is faithful, that is, when the images carry the same information as the originals — this is transinforming. Mazur derives the answer from seven theorems and reduces it to three routes to fidelity: any distortion whatever, provided it is undone by a second, opposite code; proportional distortion; and transmission without distortion — of the original itself or of an exact copy of it, which, incidentally, has to be checked more carefully than the original. According to Mazur, beyond these three ways fidelity cannot be achieved in the general case. In connection with proportionality the author also discusses scientific models and the limits of scale modelling; one of the examples is a map, which carries the same ratios of distances as the terrain even when we do not know the scale. At the end of the sequence stand two special cases in which one can establish only that something is the same, or only which of two states has occurred; according to Mazur, all knowledge begins with the first of them. Along the way the chapter discusses recognition messages, by which one checks whether the original before one is the right one.
Concepts introduced in the chapter (8): transinforming (8.1), trivial transinforming (8.2), identity transinforming (8.3), equality transinforming (8.4), analogue transinforming (8.5), compensative transinforming (8.6), comparative transinforming (8.7), eliminative transinforming (8.8).
Chapter 9. Pseudoinforming (pp. 118–140)
The chapter ends with a conclusion about losses: what was lost when several originals merged into one image cannot be restored by any reversal of the code — at most one can list what it might have been; a surplus of images of a single original, by contrast, is simply removed by keeping any one of them. In pseudoinforming no coding chain breaks off, but some converge or branch: one original gives several images, or several originals give one image. The simulative variant multiplies apparent items of information, the dissimulative one loses them, and in the confusive variant both phenomena occur at once; in its double form the confusion is complete, because every image may come from every original. To this class of phenomena Mazur assigns analysis into parts and division into classes, vagueness as a way of concealing ignorance, and ambiguous words, though he stipulates that generalisation can be useful when whole classes are compared. For this reason he prefers to speak of transformations throughout the book: the general word ‘relation’ would blur distinctions that he needs.
Concepts introduced in the chapter (9): pseudoinforming (9.1), pseudoinformation (9.2), simulative pseudoinforming (9.3), simulative pseudoinformation (9.4), dissimulative pseudoinforming (9.5), dissimulative pseudoinformation (9.6), confusive pseudoinforming (9.7), single confusive pseudoinforming (9.8), double confusive pseudoinforming (9.9).
Chapter 10. Disinforming (pp. 141–152)
In disinforming the coding chains do not intertwine, but some are broken off: an image has no original, or an original does not reach the image, so the result may be false information or its absence. The three variants are easy to show on a library catalogue: an entry with no book on the shelf (simulation; here Mazur also places literary fiction), a book with no entry in the catalogue (dissimulation), and a wrong shelf mark (the confusive variant, which covers, among other things, measurement errors, slips and distortions). An omission, Mazur notes, is harder to detect than an invention. The theorems of the chapter describe the routes back to fidelity, and the general advice is to avoid unnecessary transformations along the way.
Concepts introduced in the chapter (9): disinforming (10.1), disinformation (10.2), simulative disinforming (10.3), simulative disinformation (10.4), dissimulative disinforming (10.5), dissimulative disinformation (10.6), confusive disinforming (10.7), single confusive disinforming (10.8), double confusive disinforming (10.9).
Chapter 11. Parainforming (pp. 153–170)
The chapter goes beyond the coding chains and takes in what is commonly called meaning — an area that, according to Mazur, quantitative information theory left outside its scope. In this account the meaning of an utterance turns out to be the listener's own contribution: the words merely set off linkages that the listener already has in memory. The author takes the physical basis of parainforming from the theory of autonomous systems: the lasting trace of a flow of energy is a changed conductance, and human memory records gradually, forgets, and best retains stimuli that are frequent, strong and recent. In an organism there is also the evaluation of a stimulus from the point of view of survival, which a machine lacks — parainformation is put into a machine by its user.
Parainforming occurs when, beside ordinary transmission, additional states appear that belong to no coding chain, at the source, at the recipient or on both sides; without ordinary transmission there is no parainforming. The human example is the sentence ‘it has struck nine’, after which the listener supplies for themselves a word that nobody uttered. When the additions on both sides are the same, mutual understanding works — speech, gestures, allusions and metaphors rest on this; when they differ, variants of paradisinforming arise: seeing an allusion that was not there, failing to catch one, reading it perversely. Hence the conclusion that meaningfulness is not a property of the sentence itself but depends on the agreement of linkages on both sides: a technical text may be clear to a specialist and empty to a layperson, although it is made up of the same words.
Concepts introduced in the chapter (17): paramessage (11.1), paraoriginal (11.2), paraimage (11.3), parainformation (11.4), parainforming (11.5), paratransinforming (11.6), paratransinformation (11.7), paradisinforming (11.8), paradisinformation (11.9), simulative paradisinforming (11.10), simulative paradisinformation (11.11), dissimulative paradisinforming (11.12), dissimulative paradisinformation (11.13), confusive paradisinforming (11.14), confusive paradisinformation (11.15), single confusive paradisinforming (11.16), double confusive paradisinforming (11.17).
Chapter 12. Metainforming (pp. 171–187)
The last step of the construction is metainforming, that is, informing about informing. It is needed when what counts for control is no longer the result of a single path but the agreement or difference between the results of two: comparing them is again informing, only one level higher, and such levels can be built one above another. All the earlier kinds of informing carry over to the higher level, and the chapter gives them names with the prefix meta-. The most important conclusions concern truth and falsehood. Four theorems describe when a comparison of two paths is reliable: agreement between two results does not yet prove that they are true, because two paths distorted in the same way will give results that agree and are nonetheless false. The levels are, moreover, independent of one another. A lie differs from a falsehood not in content but in level: it is a false statement that someone additionally presents as true; with the same concepts Mazur describes the ways of reporting other people's words. Judging whether a statement is true always requires some standard, and the author explains the liar paradox as a confusion of a sentence with a statement about that sentence.
Concepts introduced in the chapter (23): metacontrol path (12.1), metamessage (12.2), metaoriginal (12.3), metaimage (12.4), metacode (12.5), trivial metacode (12.6), non-trivial metacode (12.7), metainformation (12.8), trivial metainformation (12.9), non-trivial metainformation (12.10), metainforming (12.11), simulative metainforming (12.12), dissimulative metainforming (12.13), confusive metainforming (12.14), metatransinforming (12.15), metatransinformation (12.16), metapseudoinforming (12.17), metapseudoinformation (12.18), metadisinforming (12.19), metadisinformation (12.20), metaparamessage (12.21), metaparainformation (12.22), metaparainforming (12.23).
Chapter 13. The number of items of information (pp. 188–207)
The last chapter confronts the new theory with the old one: Mazur counts individual items of information and checks how the result relates to Shannon's measure. The amount of information in bits turns out to be the number of items of identifying information, and a bit — one such item when the choice is between two possibilities. Mazur regards the widespread identification of information with its amount as a misunderstanding that stems from the theory's origins in telecommunications engineering. Counting bits makes sense where something has to be singled out from among possibilities: a textbook describes, whereas a pupil who, before an examination, guesses what they will be asked is already engaged in identification.
The result rests on a distinction between two kinds of useful information: describing information, of which as many items are needed as the chain has elements, and identifying information, whose items make it possible to point out one of the elements. The relationship between the numbers of items of each kind leads, for distinct elements, to Hartley's formula, and for classes of identical elements — to Shannon's formula.
Concepts introduced in the chapter (7): useful information (13.1), redundant information (13.2), parasitic information (13.3), describing information (13.4), reference information (13.5), reference message (13.6), identifying information (13.7).
Classification of informing (pp. 208–213)
Chapter 13 is followed by a table classifying informing, spread over two facing pages and repeated in Russian and in English (pp. 208–213). The left-hand page of the table arranges the kinds of informing by the structure of the coding chains — also for the meta- level, while the meta-meta- level is only indicated; the right-hand page gives, in the same rows, their names with an everyday explanation, from faithful to presumed informing, and holds the basic nomenclature on a single page (without the single and double variants and, on the meta- level, without the division into simulative, dissimulative and confusive variants). After the table come two summaries of the whole, a few pages each, in Russian (pp. 214–216) and in English (pp. 217–219), and an index (pp. 221–223).
Concepts introduced in the chapter: none — this part of the book contains no numbered definitions.
How to read it
It is best to begin with the introduction and Chapters 2–3, and then to read the table of explanations on p. 209 — it gives a map of the whole. The most difficult is Chapter 4: formal, with its own notation and a polemic with logic, but without it the rest does not hold together; the proofs of the successive theorems are similar to one another and can be skimmed on a first reading. Chapters 9–12 are the most accessible thanks to examples from everyday life; in Chapter 11 Mazur explicitly points to the area that quantitative information theory left outside its scope. Chapter 13 requires ease with logarithms. While reading, it is worth keeping open the definitions in order view and the table of families of terms.
Concepts from this book in the dictionary
In total: 141 entries with 141 definitions from this book — 135 in the domain qualitative information theory, 6 in the domain cybernetics.
Our own account, working version; definitions of concepts are in the dictionary, with their sources.