Semantic information

Semantic information is the semantic aspect of information, reflecting the relationship between the form of communication and its semantic content.

Starting with the works of Claude Shannon , it is commonly believed ^[1] that the concept of information consists of three aspects: syntactic , semantic and pragmatic . The syntactic is related to the technical problems of storing and transmitting information, the semantic is related to the meaning and significance of the truth of messages, the pragmatic deals with the impact of information on people's behavior. The theory of semantic information explores the field of human knowledge and is an integral part of the development of artificial intelligence ^[2] .

History

Formation of the concept of semantic information

The emergence of semiotics in the 19th century created the prerequisites for the emergence of the concept of semantic information ^[3] . It finally came about after the appearance of the Mathematical Theory of Communication , created by Claude Shannon in 1948 ^[4] . Shannon's theory, now regarded as a theory of syntactic information, completely ignores the meaning of the message. It was then that the need to create a theory of semantic information was realized.

Bar Hillel and Carnap Theory

In 1952, Yehoshua Bar-Hillel and Rudolf Carnap proposed a theory of semantic information based on the concept of logical probabilities ^[5] . The semantic information is interpreted by the authors as a synonym for semantic content, which both true and false expressions possess. Two main measures of the amount of semantic information in a sentence are considered. $s$ ${\ displaystyle s}$ $s$ . First ${\mbox{cont}}(s)$ ${\ displaystyle {\ mbox {cont}} (s)}$ ${\mbox{cont}}(s)$ defined as follows:

{\mbox{cont}}(s)=1-q(s)

{\ displaystyle {\ mbox {cont}} (s) = 1-q (s)}

{\mbox{cont}}(s)=1-q(s)

,

Where $q(s)$ ${\ displaystyle q (s)}$ $q(s)$ - absolute logical probability of the offer $s$ ${\ displaystyle s}$ $s$ . Second measure ${\mbox{inf}}(s)$ ${\ displaystyle {\ mbox {inf}} (s)}$ ${\mbox{inf}}(s)$ is a nonlinear function of the first:

{\mbox{inf}}(s)=\log _{2}{\frac {1}{1-{\mbox{cont}}(s)}}=\log _{2}{\frac {1}{q(s)}}

{\ displaystyle {\ mbox {inf}} (s) = \ log _ {2} {\ frac {1} {1 - {\ mbox {cont}} (s)}} = \ log _ {2} {\ frac {1} {q (s)}}}

{\mbox{inf}}(s)=\log _{2}{\frac {1}{1-{\mbox{cont}}(s)}}=\log _{2}{\frac {1}{q(s)}}

.

It is interesting in that for two logically independent sentences $s_{1}$ ${\ displaystyle s_ {1}}$ $s_{1}$ and $s_{2}$ ${\ displaystyle s_ {2}}$ $s_{2}$ we have the inequality: ${\mbox{cont}}(s_{1})+{\mbox{cont}}(s_{2})>{\mbox{cont}}(s_{1}\land s_{2})$ ${\ displaystyle {\ mbox {cont}} (s_ {1}) + {\ mbox {cont}} (s_ {2})> {\ mbox {cont}} (s_ {1} \ land s_ {2}) }$ ${\mbox{cont}}(s_{1})+{\mbox{cont}}(s_{2})>{\mbox{cont}}(s_{1}\land s_{2})$ where " $\land$ ${\ displaystyle \ land}$ $\land$ "- the sign of the logical connective" AND ", whereas:

{\mbox{inf}}(s_{1})+{\mbox{inf}}(s_{2})={\mbox{inf}}(s_{1}\land s_{2})

{\ displaystyle {\ mbox {inf}} (s_ {1}) + {\ mbox {inf}} (s_ {2}) = {\ mbox {inf}} (s_ {1} \ land s_ {2}) }

, (*)

which is more suitable for measuring the amount of information.

To determine the logical probabilities of sentences, Bar-Hillel and Karnap construct a formal language and use it to describe all kinds of states of the universe (the so-called " many possible worlds "). We give an example of a simple language in which there is one constant $a$ ${\ displaystyle a}$ (by her we mean the girl Alice) and two predicates : $B$ ${\ displaystyle B}$ and $W$ ${\ displaystyle W}$ , denoting the properties of "beautiful" and "smart." Then the expression $B(a)$ ${\ displaystyle B (a)}$ means the sentence "Alice is beautiful," and the expression $W(a)$ ${\ displaystyle W (a)}$ “Alice is smart.” Now we use the logical connective “NOT”, which we denote by the symbol: “ $\neg$ ${\ displaystyle \ neg}$ ". Then the expression $\neg B(a)$ ${\ displaystyle \ neg B (a)}$ will mean the sentence "Alice is not beautiful," but the expression $\neg W(a)$ ${\ displaystyle \ neg W (a)}$ “Alice is not smart.” Now we can compile all possible descriptions of the states of the universe for our humble language. There will be four of them.

B(a)\land W(a)

{\ displaystyle B (a) \ land W (a)}

B(a)\land \neg W(a)

{\ displaystyle B (a) \ land \ neg W (a)}

\neg B(a)\land W(a)

{\ displaystyle \ neg B (a) \ land W (a)}

\neg B(a)\land \neg W(a)

{\ displaystyle \ neg B (a) \ land \ neg W (a)}

As you can see, each world of the universe consists of logically independent atomic sentences (and their negations), called basic. Typically, formal languages use many constants and many predicates, and not necessarily single ones . So the number of worlds can be very large.

If no preconditions are given, then the logical probabilities of all worlds are the same. In this case, the absolute logical probability of the proposal $s$ ${\ displaystyle s}$ equal to the ratio of the number of worlds in which $s$ ${\ displaystyle s}$ verily, to the total number of worlds in the universe. In the Bar-Hillel and Karnap theory, the logical probabilities of analytic expressions are the same and equal to unity (since they are true in all worlds), and the logical probability of a contradiction is zero. The logical probabilities of synthetic expressions are in the range from zero to unity.

The more worlds in the universe, the greater the uncertainty (as to which world is true). After receiving the message $s$ ${\ displaystyle s}$ uncertainty decreases as those worlds in which $s$ ${\ displaystyle s}$ false, can be excluded from consideration. Semantic information in a sentence $s$ ${\ displaystyle s}$ understood as the set of excluded worlds (it is indicated by the symbol ${\mbox{Cont}}(s)$ ${\ displaystyle {\ mbox {Cont}} (s)}$ ) Regarding this definition, the authors write that it is consistent with the ancient philosophical principle of " omnis determinatio est negatio " (" every definition is an exception "). Now for the measure ${\mbox{cont}}(s)$ ${\ displaystyle {\ mbox {cont}} (s)}$ we can write:

{\mbox{cont}}(s)={\frac {|{\mbox{Cont}}(s)|}{|{\mbox{U}}|}}

{\ displaystyle {\ mbox {cont}} (s) = {\ frac {| {\ mbox {Cont}} (s) |} {| {\ mbox {U}} |}}}

,

Where $|{\mbox{Cont}}(s)|$ ${\ displaystyle | {\ mbox {Cont}} (s) |}$ Is the power of the set ${\mbox{Cont}}(s)$ ${\ displaystyle {\ mbox {Cont}} (s)}$ , $|{\mbox{U}}|$ ${\ displaystyle | {\ mbox {U}} |}$ - the power of many worlds of the universe ${\mbox{U}}$ ${\ displaystyle {\ mbox {U}}}$ .

The amount of semantic information in the message $s$ ${\ displaystyle s}$ regarding recipient knowledge $e$ ${\ displaystyle e}$ defined as follows:

{\mbox{inf}}(s/e)={\mbox{inf}}(s\land e)-{\mbox{inf}}(e)=\log _{2}{\frac {q(e)}{q(s\land e)}}=\log _{2}{\frac {1}{q(s/e)}}

{\ displaystyle {\ mbox {inf}} (s / e) = {\ mbox {inf}} (s \ land e) - {\ mbox {inf}} (e) = \ log _ {2} {\ frac {q (e)} {q (s \ land e)}} = \ log _ {2} {\ frac {1} {q (s / e)}}}

,

Where $q(s/e)$ ${\ displaystyle q (s / e)}$ - relative (conditional) logical probability of the truth of the statement $s$ ${\ displaystyle s}$ subject to the truth of the expression $e$ ${\ displaystyle e}$ .

Remarkably, the formulas of the Bar-Hillel and Karnap theory are similar in appearance to the formulas of the Shannon theory. Both there and here we have logarithms and probabilities . Only Shannon has all the probabilities - statistical (i.e. empirical ), not logical.

If the logical probability of expression $s\land e$ ${\ displaystyle s \ land e}$ less logical probability of expression $e$ ${\ displaystyle e}$ then message $s$ ${\ displaystyle s}$ brings new information to the recipient, thus enriching his knowledge. If a $e$ ${\ displaystyle e}$ implicates $s$ ${\ displaystyle s}$ then $s\land e$ ${\ displaystyle s \ land e}$ equivalently $e$ ${\ displaystyle e}$ and message $s$ ${\ displaystyle s}$ does not carry information to the addressee (since there is nothing new for him). If the expression $s\land e$ ${\ displaystyle s \ land e}$ is a contradiction then $q(s\land e)=0$ ${\ displaystyle q (s \ land e) = 0}$ . The amount of semantic information in the contradiction according to Bar-Hillel and Karnap is equal to infinity . This paradoxical result subsequently led to criticism from Luciano Floridi.

Alternative Ideas

Although the theory of Bar-Hillel and Karnap still attracts the attention of researchers, it caused a flood of new ideas. Alexander Kharkevich proposed measuring the value of information on the change in the probability of achieving a specific goal arising under the influence of this message ^[6] . Julius Schreider believed that the amount of semantic information in a message of any nature can be estimated as the degree of change in the recipient’s knowledge system as a result of message perception ^[7] . The idea of the semantic aspect of the connection between information and entropy was first proposed in 1966 by the Soviet philosopher and logician Evgeny Kazimirovich Voishvillo in his work “ An attempt to semantic interpretation of statistical concepts of information and entropy ”.

Modern Theories of Semantic Information

Floridi Theory

In his 2004 work, Luciano Floridi attacks the theory of Bar Hillel and Carnap from the first line: “ The Triangle has four sides”: according to the classical theory of semantic information, this contradiction contains more semantic content than the conditionally true statement “The Earth has only one Moon “ ” ^[8] . Floridi called it the " Bar-Hillel-Carnap paradox ." He sees the solution to this paradox in the fact that the amount of semantic information in messages should depend not only on the semantic content contained in them, but also on the significance of the truth of these messages. Floridi introduced the concept of a conditionally false sentence ( contingently false sentence ), which is a conjunction of two of its components, one of which is true, and the second is false. An example of such a sentence is the saying: "The moon rotates around the Earth and inside it is hollow." Such a proposal simultaneously carries information (for those who do not know that the Moon revolves around the Earth) and misinformation (in everyday life it is often necessary to encounter something like this - it is easier to promote disinformation if it is supplemented by some amount of information).

From the point of view of classical logic, a conditionally false sentence is simply false and carries only misinformation. However, the above example shows that this is actually not the case. The original theory of Bar-Hillel and Karnap is not able to solve this antinomy . Therefore, Floridi rejected it (as a "weak" theory) and created his own - "strong". He refused to use logical probabilities and stated that the theory of semantic information should not be similar to Shannon's theory ^[9] . In his own interpretation, the amount of semantic information in a message is determined by the degree to which this message matches the situation (that is, what is happening in a given place and at a given time). The mismatch arises either as a result of the message being empty or as a result of its inaccuracy. In his theory, Floridi does not directly use the concept of misinformation; instead, he introduces the concept of the degree of inaccuracy of conditionally false sentences. The degree of inaccuracy in a conditionally false sentence $s$ ${\ displaystyle s}$ is equal to:

-v(s)=-{\frac {f(s)}{l(s)}}

{\ displaystyle -v (s) = - {\ frac {f (s)} {l (s)}}}

,

Where $f(s)$ ${\ displaystyle f (s)}$ - the number of false atomic expressions in $s$ ${\ displaystyle s}$ ; $l(s)$ ${\ displaystyle l (s)}$ - the total number of atomic sentences in $s$ ${\ displaystyle s}$ . To determine the truth of atomic sentences, it is necessary to accept the principle of a priori omniscience. The degree of emptyness of the true sentence $s$ ${\ displaystyle s}$ calculated by the formula:

+v(s)={\frac {m(s)}{n}}

{\ displaystyle + v (s) = {\ frac {m (s)} {n}}}

,

Where $m(s)$ ${\ displaystyle m (s)}$ - the number of worlds of the universe in which $s$ ${\ displaystyle s}$ truly; $n$ ${\ displaystyle n}$ Is the total number of worlds of the universe (note that, according to this definition, the quantity $+v(s)$ ${\ displaystyle + v (s)}$ exactly equal to the value of logical probability $q(s)$ ${\ displaystyle q (s)}$ ) Further, Floridi introduces the concept of the function of the degree of informational content:

i(s)=1-v^{2}(s)

{\ displaystyle i (s) = 1-v ^ {2} (s)}

.

Amount of semantic information $i^{*}(s)$ ${\ displaystyle i ^ {*} (s)}$ in the message $s$ ${\ displaystyle s}$ equal to a certain integral of the function of the degree of informativeness $i(s)$ ${\ displaystyle i (s)}$ :

i^{*}(s)={\frac {3}{2}}\int \limits _{v(s)}^{1}(1-x^{2})\mathrm {d} x=1-{\frac {3v(s)}{2}}+{\frac {v^{3}(s)}{2}}

{\ displaystyle i ^ {*} (s) = {\ frac {3} {2}} \ int \ limits _ {v (s)} ^ {1} (1-x ^ {2}) \ mathrm {d } x = 1 - {\ frac {3v (s)} {2}} + {\ frac {v ^ {3} (s)} {2}}}

.

Despite all the differences between the classical theory and the theory of Floridi, they have something in common. If a $s$ ${\ displaystyle s}$ is a true proposition, then the quantity $+v(s)$ ${\ displaystyle + v (s)}$ equal to the value of logical probability $q(s)$ ${\ displaystyle q (s)}$ . Measure $i^{*}(s)$ ${\ displaystyle i ^ {*} (s)}$ similar measure ${\mbox{cont}}(s)$ ${\ displaystyle {\ mbox {cont}} (s)}$ , but unlike the latter, is a nonlinear function $v(s)$ ${\ displaystyle v (s)}$ . Unfortunately, in Floridi's theory there is nothing like a measure ${\mbox{inf}}(s)$ ${\ displaystyle {\ mbox {inf}} (s)}$ possessing a remarkable property (*) for logically independent sentences.

Theory of Semantic Information and Disinformation

The problem raised by Floridi can be solved within the framework of a theory based on logical probabilities. It should be noted that by the beginning of this century, some scientists had formed a skeptical attitude towards the inductive logic of Karnap ^[10] . However, modern mathematicians were able to change the situation by modifying this theory ^[11] ^[12] ^[13] . Thanks to this, interest in logical probabilities has revived again.

In ^{[14], it is} proposed to modify the classical theory of semantic information by including the concept of misinformation carried by a false message. In the new theory, as in Floridi's theory, many different situations (points of space-time) are considered. The same sentence of a language can be true in one situation and false in another. Since the recipient of messages cannot be insured against errors in assessing their truthfulness, the amount of semantic information is evaluated separately from the point of view of the recipient and from the point of view of an omniscient expert.

In each specific situation, the true message carries only information, and absolutely false - only misinformation. Conditionally false offer $s$ ${\ displaystyle s}$ regarded as a conjunction : $s_{T}\land s_{F}$ ${\ displaystyle s_ {T} \ land s_ {F}}$ where $s_{T}$ ${\ displaystyle s_ {T}}$ - the true part of the message, $s_{F}$ ${\ displaystyle s_ {F}}$ - the false part of the message. Moreover, it is required that $s_{T}$ ${\ displaystyle s_ {T}}$ and $s_{F}$ ${\ displaystyle s_ {F}}$ were logically independent (this is necessary, in particular, so that the contradiction does not turn out to be a conditionally false proposal). Then abnormal measures of the amount of information ${\mbox{in}}_{E}(s)$ ${\ displaystyle {\ mbox {in}} _ {E} (s)}$ and the amount of misinformation ${\mbox{mi}}_{E}(s)$ ${\ displaystyle {\ mbox {mi}} _ {E} (s)}$ in a conditionally false sentence $s$ ${\ displaystyle s}$ from the point of view of an expert are defined as follows:

{\mbox{in}}_{E}(s)={\mbox{cont}}(s_{T})

{\ displaystyle {\ mbox {in}} _ {E} (s) = {\ mbox {cont}} (s_ {T})}

,

{\mbox{mi}}_{E}(s)={\mbox{cont}}(s_{F})

{\ displaystyle {\ mbox {mi}} _ {E} (s) = {\ mbox {cont}} (s_ {F})}

.

Index " $E$ ${\ displaystyle E}$ "That marks the characters" ${\mbox{in}}$ ${\ displaystyle {\ mbox {in}}}$ "And" ${\mbox{mi}}$ ${\ displaystyle {\ mbox {mi}}}$ ”In the formulas, indicates that the amounts of information and misinformation are considered from the expert’s point of view. Normalized measures of the amount of semantic information ${\mbox{inf}}_{E}(s)$ ${\ displaystyle {\ mbox {inf}} _ {E} (s)}$ and misinformation ${\mbox{mis}}_{E}(s)$ ${\ displaystyle {\ mbox {mis}} _ {E} (s)}$ in a conditionally false sentence $s$ ${\ displaystyle s}$ from an expert point of view:

{\mbox{inf}}_{E}(s)=\log _{2}{\frac {1}{1-{\mbox{cont}}(s_{T})}}=\log _{2}{\frac {1}{q(s_{T})}}

{\ displaystyle {\ mbox {inf}} _ {E} (s) = \ log _ {2} {\ frac {1} {1 - {\ mbox {cont}} (s_ {T})}} = \ log _ {2} {\ frac {1} {q (s_ {T})}}}

,

{\mbox{mis}}_{E}(s)=\log _{2}{\frac {1}{1-{\mbox{cont}}(s_{F})}}=\log _{2}{\frac {1}{q(s_{F})}}

{\ displaystyle {\ mbox {mis}} _ {E} (s) = \ log _ {2} {\ frac {1} {1 - {\ mbox {cont}} (s_ {F})}} = \ log _ {2} {\ frac {1} {q (s_ {F})}}}

.

The contradiction from the point of view of the expert carries a zero amount of information and an infinite amount of misinformation. Thus, the Bar-Hillel-Carnap paradox is solved. The endless amount of misinformation is explained by the fact that if the contradiction suddenly seemed to someone to be true, then the world would change for him beyond recognition. Two words can’t describe it. Suppose that the recipient of information has conditionally false knowledge $e$ ${\ displaystyle e}$ equivalent conjunctions: $e_{T}\land e_{F}$ ${\ displaystyle e_ {T} \ land e_ {F}}$ where $e_{T}$ ${\ displaystyle e_ {T}}$ - the true part of his knowledge, $e_{F}$ ${\ displaystyle e_ {F}}$ - misconceptions. Then from the expert’s point of view, having received a conditionally false message $s$ ${\ displaystyle s}$ , the addressee really has semantic information and misinformation in the following quantities:

{\mbox{inf}}_{E}(s/e)=\log _{2}{\frac {q(e_{T})}{q(s_{T}\land e_{T})}}=\log _{2}{\frac {1}{q(s_{T}/e_{T})}}

{\ displaystyle {\ mbox {inf}} _ {E} (s / e) = \ log _ {2} {\ frac {q (e_ {T})} {q (s_ {T} \ land e_ {T })}} = \ log _ {2} {\ frac {1} {q (s_ {T} / e_ {T})}}}

,

{\mbox{mis}}_{E}(s/e)=\log _{2}{\frac {q(e_{F})}{q(s_{F}\land e_{F})}}=\log _{2}{\frac {1}{q(s_{F}/e_{F})}}

{\ displaystyle {\ mbox {mis}} _ {E} (s / e) = \ log _ {2} {\ frac {q (e_ {F})} {q (s_ {F} \ land e_ {F })}} = \ log _ {2} {\ frac {1} {q (s_ {F} / e_ {F})}}}

.

If the recipient perceives $s$ ${\ displaystyle s}$ like a true sentence and conjunction $s\land e$ ${\ displaystyle s \ land e}$ is not a contradiction, then from his point of view he received the following amount of information:

{\mbox{inf}}_{R}(s/e)=\log _{2}{\frac {1}{q(s/e)}}={\mbox{inf}}_{E}(s/e)+{\mbox{mis}}_{E}(s/e)

{\ displaystyle {\ mbox {inf}} _ {R} (s / e) = \ log _ {2} {\ frac {1} {q (s / e)}} = {\ mbox {inf}} _ {E} (s / e) + {\ mbox {mis}} _ {E} (s / e)}

.

Index " $R$ ${\ displaystyle R}$ "Indicates the addressee’s rating. It is obvious that only an expert can determine the exact amount of information (and misinformation) in a received message, and the recipient is capable of only more or less accurate estimates.

Theory of Universal Semantic Information

A formal description of semantic information applicable to all types of physical systems (living and nonliving) was given by mathematician David Wolpert in his work "Semantic information, agency, and nonequilibrium statistical physics": the syntactic information that a physical system has about the environment, and which is casually necessary for the system to maintain its own existence in a state of low entropy.

Casual need is defined in terms of counter-factual interventions that randomize correlations between the system and the environment. The criterion for the degree of autonomy of a physical system is the amount of semantic information available.

Notes

↑ Shannon CE, Weaver W., (1949), The Mathematical Theory of Communication, Urbana: University of Illinois Press. Foreword by Richard E. Blahut and Bruce Hajek; reprinted in 1998.
↑ Luger D.F. Artificial intelligence: strategies and methods for solving complex problems. - M.: Williams Publishing House, 2005. - 864 p. ISBN 5-8459-0437-4 (Russian)
↑ Dmitriev V.I. Applied Information Theory. - M .: Higher School, 1989 .-- 320 p. ISBN 5-06-000038-9
↑ Shannon CE, (1948), A Mathematical Theory of Communication. Bell syst. Tech. J., 27: 379-423, 623-656.
↑ Bar-Hillel Y., Carnap R., (1952), «An Outline of a Theory of Semantic Information», Technical Report No. 247, October 27, Research Laboratory of Electronics. – 49. [1] Архивировано 12 июля 2013 года.
↑ Харкевич А. А. О ценности информации, «Проблемы кибернетики», 1960, в. 4. – с. 54.
↑ Шрейдер Ю. А., (1965), Об одной модели семантической теории информации, «Проблемы кибернетики», в. 13. – с. 233-240.
↑ Floridi L. (2004), «Outline of a Theory of Strongly Semantic Information», Minds and Machines, 14(2), 197-222. [2] Архивная копия от 2 августа 2014 на Wayback Machine
↑ Floridi L. (2011), Semantic Conception of Information, In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta, [3]
↑ Hajek Alan. (2007). Interpretation of probability. In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta, [4] (недоступная ссылка)
↑ Maher Patrick, (2010). Explication of Inductive Probability. Journal of Philosophical Logic 39 (6): 593-616.
↑ Zabell SI (2004). Carnap and the Logic of Inductive Inference. In Dov M. Gabbay, John Woods & Akihiro Kanamori (eds.), Handbook of the History of Logic. Elsevier 265-309.
↑ Ruurik Holm (2013). Non-Zero Probabilities for Universal Generalizations. Synthese 190 (18): 4001-4007.
↑ Погорелов О. А. (2015). Семантическая информация и дезинформация //Сборник научных статей по итогам V Международной научно-практической конференции «Информатика, Математическое моделирование, Экономика» (г. Смоленск, 11-15 мая 2015 г.), с. 132-143. [five]

[list1-1] Shannon CE, Weaver W., (1949), The Mathematical Theory of Communication, Urbana: University of Illinois Press. Foreword by Richard E. Blahut and Bruce Hajek; reprinted in 1998.

[list2-2] Luger D.F. Artificial intelligence: strategies and methods for solving complex problems. - M.: Williams Publishing House, 2005. - 864 p. ISBN 5-8459-0437-4 (Russian)

[list3-3] Dmitriev V.I. Applied Information Theory. - M .: Higher School, 1989 .-- 320 p. ISBN 5-06-000038-9

[list4-4] Shannon CE, (1948), A Mathematical Theory of Communication. Bell syst. Tech. J., 27: 379-423, 623-656.

[list5-5] Bar-Hillel Y., Carnap R., (1952), «An Outline of a Theory of Semantic Information», Technical Report No. 247, October 27, Research Laboratory of Electronics. – 49. [1] Архивировано 12 июля 2013 года.

[list6-6] Харкевич А. А. О ценности информации, «Проблемы кибернетики», 1960, в. 4. – с. 54.

[list7-7] Шрейдер Ю. А., (1965), Об одной модели семантической теории информации, «Проблемы кибернетики», в. 13. – с. 233-240.

[list8-8] Floridi L. (2004), «Outline of a Theory of Strongly Semantic Information», Minds and Machines, 14(2), 197-222. [2] Архивная копия от 2 августа 2014 на Wayback Machine

[list9-9] Floridi L. (2011), Semantic Conception of Information, In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta, [3]

[list10-10] Hajek Alan. (2007). Interpretation of probability. In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta, [4] (недоступная ссылка)

[list11-11] Maher Patrick, (2010). Explication of Inductive Probability. Journal of Philosophical Logic 39 (6): 593-616.

[list12-12] Zabell SI (2004). Carnap and the Logic of Inductive Inference. In Dov M. Gabbay, John Woods & Akihiro Kanamori (eds.), Handbook of the History of Logic. Elsevier 265-309.

[list13-13] Ruurik Holm (2013). Non-Zero Probabilities for Universal Generalizations. Synthese 190 (18): 4001-4007.

[list14-14] Погорелов О. А. (2015). Семантическая информация и дезинформация //Сборник научных статей по итогам V Международной научно-практической конференции «Информатика, Математическое моделирование, Экономика» (г. Смоленск, 11-15 мая 2015 г.), с. 132-143. [five]