Jakościowa teoria informacji (‘A Qualitative Theory of Information’, 1970) has the structure of a rigorous exposition: the concepts of the theory receive numbered definitions here (apart from a few auxiliary ones, discussed at the end), and every definition uses only concepts defined earlier or adopted in the text without definition. This guide follows the same sequence, from the simplest concepts to the most complex ones, and explains why each concept is introduced. The definitions are found on the entry pages to which the links lead; the full list in the order of the book is given in the definitions in order view, and the statistics for the domain on the Qualitative information theory page.
Method: phenomena first, then the name
Mazur does not begin by asking what people call information. Such a procedure would assign a meaning to the word before that meaning had been established. Instead he chooses an object of study: the phenomena of control, because in every situation in which someone speaks of information, what is at stake is striving towards some goal. The name ‘information’ will be given later, to one of the concepts singled out in the analysis, by terminological convention (p. 27).
That is why, after the introductory chapter, the numbered exposition opens with the concepts of cybernetics: the controlled system, the controlling system, the control path, the control loop and the control process. In every path one system acts on another, so Mazur considers the path in general, between a source of action and a receiver of action.
Messages and sets of messages
In control, what matters is not forces and amounts of energy but differences between physical states. This leads to the theory's own fundamental concept: the message, that is, a physical state in the path that counts because it can be distinguished from another. A message can be anything that can be distinguished: a measured value, a colour, a letter, a tone, a road sign (p. 38).
Messages can be grouped in two ways. A transverse set of messages consists of the messages at one point of the path; a longitudinal set of messages consists of the messages arising one from another in succession along the path. The message at the output of the source is the original, the message at the input of the receiver is the image, and a message in between is an intermessage. Mazur gives many examples: the values of a measured quantity and the readings of an instrument, the text of a telegram as sent and as received, terrain and a map, French words and their English equivalents in a dictionary (Jakościowa teoria informacji, p. 36). Originals need not precede images: a letter about a past event or the fossil of an extinct species is control on the basis of history, the colour of a flame telling of its temperature is control on the basis of diagnosis, and the present position of celestial bodies announcing a future eclipse is control on the basis of prognosis (pp. 37–38).
Transformation: the link between messages
Once there are messages, the links between them have to be described. Here Mazur deliberately gives up the concept of relation, which would require too many qualifications, and builds a tool of his own (p. 41). The basis is the association of messages — an unordered pair — and the transformation, that is, the passage from one message of the pair to the other. The message from which one starts is the primary message, the result — the secondary message.
On this pair of concepts Mazur builds the whole classification. A transformation is a non-trivial transformation when the messages differ, and a trivial transformation when they do not; a trivial one may be an identity transformation (one and the same message) or an equality transformation (two separate but identical messages). Every transformation has its inverse transformation. A transformation consists of operations, and every operation has its kind of operation and its parameter of operation: in adding six, the kind is addition and the parameter is the number six (p. 45). Hence the operational transformation, determined by operations; the principal transformation, one for all the pairs of a given set; and the associative transformation, determined only by the fact that it links a given pair — tables and bilingual dictionaries rest on such links (p. 64). Pairs arranged in a sequence form a transformation chain, and the composition of its successive transformations is the resultant transformation.
Code and information
Now the same material is used twice. A pair taken from a longitudinal set is a coding association, and the transformation in it is a code. A pair taken from a transverse set is an informational association, and the transformation in it is information. A code thus describes the passage along the path, information — the link between messages at one point of the path. Information here is not a quantity but a transformation.
The whole classification of transformations carries over to both concepts: there are trivial codes and non-trivial codes, identity codes, equality codes, inverse codes, operational codes, principal codes, associative codes, and likewise trivial information, non-trivial information, operational information or principal information. Mazur's examples: a letter delivered to its addressee is an identity code, a copy of it — an equality code; using a map in the field is the inverse code of the one by which the map was drawn; an English–French dictionary contains the inverse codes of a French–English dictionary (p. 61). The passage from most (Polish for ‘bridge’) to mosty (‘bridges’) is operational information: the kind of operation is appending, the parameter is a letter (p. 75). The messages of one longitudinal set, linked by successive codes, form a coding chain, and the messages of one transverse set, linked by items of information, form an informational chain.
Informing and its four main varieties
Informing consists in the information present on the side of the originals being converted into information on the side of the images — as in measurement or in drawing a map (p. 82). Comparing the number of items of non-trivial information on the two sides gives three cases: simulative informing (the images have more of them), dissimulative informing (fewer) and confusive informing (both at once). The fundamental division, however, follows the structure of the coding chains.
Transinforming occurs when the images carry the same information as the originals. Trivial transinforming is the use of the originals themselves — which is why offices demand original documents (identity transinforming) — or of faithful copies and reproductions (equality transinforming, pp. 94–95). Analogue transinforming is a map of terrain or an enlargement of a photograph (p. 100); compensative transinforming — a voltmeter whose deflection grows with the square of the voltage, while the scale is spaced so that the error cancels out (pp. 108–109); comparative transinforming — weighing on a pan balance, where what counts is the zero position (p. 113); eliminative transinforming rests on the separateness of coding chains.
Pseudoinforming occurs when certain messages belong to several coding chains at once; its result is pseudoinformation. When the original is shared and the images differ, we have simulative pseudoinforming — like a prism splitting white light into colours (p. 120). When the image is shared while the originals differ — dissimulative pseudoinforming: a burnt-out signal lamp stays dark regardless of whether there is voltage or not (p. 122). Combining the two gives confusive pseudoinforming.
Disinforming is the case in which some coding chains are incomplete; its result is disinformation. If a chain lacks an original, the variety is simulative disinforming: a damaged voltmeter drops to zero although the voltage has not changed, a timetable lists a train that does not run (p. 142). If it lacks an image — dissimulative disinforming: the pointer stays at zero even though voltage has appeared (p. 143). A measurement error, when an ammeter shows a value other than the real one, is confusive disinforming (p. 145).
The fourth variety is introduced by the paramessage: a message that belongs to the set compared at one point of the path, but lies on no coding chain: no code carries it along the path, although it appears thanks to another message at the same point. Information in a pair containing such a message is parainformation, and informing in which it occurs is parainforming. Mazur illustrates it with signal lamps at a power station and at a substation: the lighting of the lamp at the substation tells the staff that the lamp at the power station has lit up, although nothing carries the glow of one lamp to the other: each lights up by itself from the voltage at its own end of the line, so both are paramessages (pp. 157 and 167). This is paratransinforming; its opposite is paradisinforming. According to Mazur, paratransinforming plays the greatest role in communication between people (p. 158).
The meta level
Chapter twelve adds a new level: the metacontrol path, that is, a path of a higher order which controls by comparing what various ordinary paths have brought (p. 177). Its material is metamessages, from which come metaoriginals and metaimages; the passage between them is a metacode, and metainformation and metainforming reproduce the structure of information and informing at a higher level. That is why the whole family of varieties returns, from metatransinforming to metadisinforming, and on to metaparainforming. Mazur reaches this level by comparing the readings of different paths, and the limit of such comparison is shown by the example of two watches that are slow by the same amount: they agree, and yet both are wrong, because when two readings agree, their error cannot be detected by comparing one with the other (p. 176).
The number of items of information
The last chapter returns to the question of quantity. Among the items of information in a chain, Mazur distinguishes useful information for a given process, redundant information, which follows from other information, and parasitic information, coming from outside the process. Determining any message of a chain requires describing information, including the reference information that links the first message with the reference message: in an arithmetic progression two items suffice — the first term and the common difference (p. 192). Singling out one distinguished message is the job of identifying information, and only the number of these items corresponds to the classical amount of information. The amount of information therefore makes sense only where the point is to identify a message; the number of items of describing information always makes sense — Mazur shows this with the dependence of a circle's diameter on its radius (p. 205).
Most of the domain's terms (106 of the 135) grow out of six bases and prefixes; the table of these compounds is shown in the families of terms view.
Open questions in the dictionary and the theory
In this domain the dictionary graph shows neither concepts without an entry nor entries without links: every concept that the graph recognises in the definitions has an entry, apart from the set, which 34 of the 135 entries use and which Mazur employs without definition. The definitions themselves do, however, reveal areas for further work:
- compound transformation (and compound information) occurs in seven definitions of the chapter on metainforming (and in the definition of resultant transformation in Chapter 4), yet has no entry of its own in the dictionary.
- metacoding chain (definitions 12.17, 12.19, 12.21) and metainformational chain (12.21) are counterparts of the coding chain and the informational chain, but have no definitions of their own.
- single-operation transformation is used in the definitions of inverse operation and of analogue transinforming without a separate entry.
- the order of exposition is kept in this domain without exception: every numbered definition uses only concepts defined earlier or adopted without definition (checked against the content of all 141 definitions).
- the table of families of terms has 42 empty cells — compounds of a prefix with a base that Mazur did not define; each of them is a question of whether such a concept makes sense and what it would mean.