Professor emeritus, Opole University, Poland, affiliated collaborator of Cardinal Stefan Wyszyński University in Warsaw. Interests: mathematical logic, logic and philosophy of language, information science.
Logical and philosophical literature provides different classifications of reasoning. In the Poli... more Logical and philosophical literature provides different classifications of reasoning. In the Polish literature on the subject, for instance, there are three popular ones accepted by representatives of the Lvov-Warsaw School: Jan Łukasiewicz, Tadeusz Czeżowski and Kazimierz Ajdukiewicz (Ajdukiewicz in Logika pragmatyczna [Pragmatic Logic]. PWN, Warsaw (1965, 2nd ed. 1974). Translated as: Pragmatic Logic. Reidel & PWN, Dordrecht, 1975). The author of this paper, having modified those classifications, distinguished the following types of reasoning: (1) deductive and (2) non-deductive, and additionally two types of them in each of the two, depending on the manner of combining their premises with the conclusion through the relation of classical logical entailment. Consequently, the four types of reasoning: unilateral deductive (incl. its sub-types: deductive inference and proof), bilateral deductive (incl. complete induction), and reductive (incl. the sub-types: explanation and verificat...
The paper is an attempt at a logical explication of some crucial notions of current general seman... more The paper is an attempt at a logical explication of some crucial notions of current general semantics and pragmatics. A general, axiomatic, formal-logical theory of meaning and interpretation is outlined in this paper. In the theory, according to the tokentype distinction of Peirce, language is formalised on two levels: first as a language of tokenobjects (understood as material, empirical, enduring through time-and space objects) and then-as a language of type-objects (understood as abstract objects, as classes of tokens). The basic concepts of the theory, i.e. the notions: meaning, denotation and interpretation of well-formed expressions (wfes) of the language are formalised on the type-level, by utilising some semantic-pragmatic primitive notions introduced on the token-level. The paper is divided into two parts. In Part I a theory of meaning and denotation is proposed, and in Part II-its expansion to the theory of meaning and interpretation is presented. The meaning a wfe is defined as an equivalence class of the relation possessing the same manner of using types (cf. Ajdukiewicz [1934], Wittgenstein [1953]). The concept of denotation is defined by means of the relation of referring which holds between wfe-types and objects of reality described by the given language.
The Warsaw School of Logic (WSL) was the famous branch of the Lvov-Warsaw School (LWS) – the most... more The Warsaw School of Logic (WSL) was the famous branch of the Lvov-Warsaw School (LWS) – the most important movement in the history of Polish philosophy. Logic made the most important field in the activities of the WSL. The aim of this work is to highlight the role and significance of the WSL in the history of logic in the 20th century. Keywords: history of logic, program assumptions and ideas of the WSL, Łukasiewicz, Leśniewski, Tarski, major achievements of representative of the WSL, importance of the WSL in the history of logic in the 20th century.
STUDIES IN LOGIC, GRAMMAR AND RHETORIC 20 (33) 2010, 2010
The considerations presented in this work are an attempt at giving an
answer to the arising doubt... more The considerations presented in this work are an attempt at giving an answer to the arising doubts: it is obvious to philosophers and logicians that such considerations must be grounded on a relevant conception of the truth and the lie, on bringing up one of the most difficult and disturbing philosophical problems, that is the problemate of the truth, on investigating what the lie is. The confusion about the notions related to the ambiguous terms of “the truth” and “the lie” introduces, in turn, a confusion connected with attempts at answering the questions posed. Thus, in the first part of this paper, we will deal with the very notion itself, or – more precisely – with the notions of the truth; in the second one– with the notions of the lie, and in the third part – we will juxtapose the notions of the truth and the lie in such a way that in each case it should be possible to provide an answer to the question asked in the title of the work. Part four, being the final one, contains certain summary of it, as well as final considerations as a peculiar challenge.
This article explores the question of how the members of the Lvov-Warsaw School promoted values t... more This article explores the question of how the members of the Lvov-Warsaw School promoted values that can be regarded as components of so-called logical culture. The author argues that these values are strictly connected with science. With references to Łukasiewicz, Czeżowski, and Kotarbiński, the article explores how values shape the logical culture and determines society as directed towards values. The article connects the meta-philosophical perspective with the philosophical one.
Lata sześćdziesiąte przyniosły z jednej strony akceptację zjawiska nieostrości w języku i nauce, ... more Lata sześćdziesiąte przyniosły z jednej strony akceptację zjawiska nieostrości w języku i nauce, z drugiej — nowe, zorientowane semantycznie lub informatycznie, rozwiązania samego problemu nieostrości. Rezultaty tych rozwiązań
okazały się jednak nie dość zadowalające. Świadczą o tym chociażby wciąż nowe
próby i poszukiwania w tym zakresie, prowadzone w różnych ośrodkach, w tym w
Polsce (Z. Muszyński (red.) 1988). Prekursorem formalnego podejścia do problemu
nieostrości i w ogóle badań nad nieostrością w Polsce jest logik — T. Kubiński (1958),
który zbudował nieklasyczny rachunek nazw nieostrych. W tym miejscu należy też
zaznaczyć, że w 1982 roku powstała formalna teoria komplementarna w stosunku do
teorii Zadeha — teoria zbiorów przybliżonych. Zbudował ją znany polski informatyk
— Z. Pawlak (1982, 1991).
Teoria Pawlaka podchodzi do zagadnienia nieostrości nienumerycznie, w przeciwieństwie do ilościowej charakterystyki zjawiska nieostrości przez Zadeha. Porusza ona aspekty jakościowe i oparta jest na idei zbioru aproksymowanego przez parę zbiorów zwanych d o l n y m i g ó r n y m p r z y b l i ż e n i e m , z b i o r u . Przybliżenia te określają, odpowiednio, pozytywny i negatywny zakres pojęcia nieostrego. Podejście Pawlaka wiąże się z odniesieniem do pojęcia wiedzy podmiotu poznającego o obiektach badanej rzeczywistości (por. Pawlak 1992). Ma więc ono jednocześnie charakter epistemiczny. Wiedza owa jest determinowana przez układ pojęć, a ten z kolei — przez system ich zakresów. Gdy zakres pojęcia jest nieostry, wyznacza go zbiór przybliżony, rozumiany jako rodzina zbiorów mających te same przybliżenia dolne i górne. Zasadnicze idee koncepcji Pawlaka miały ogromny wpływ na zbudowanie przeze mnie logicznej teorii pojęć nieostrych, której zarys jest przedstawiony w dalszej części tej pracy, przy czym inspiracji dostarczyły mi także pomysły i rozwiązania innych badaczy wymienionych w tej części prac
An attempt at reaching the past, extracting the essence of humanity, creative achievements and ma... more An attempt at reaching the past, extracting the essence of humanity, creative achievements and making them real from already partially obliterated traces is not an easy task from the personal perspective.
The Paideia Archive: Twentieth World Congress of Philosophy, 1998
This paper sketches or signals some ideas, results, and proposals connected with the theoretical ... more This paper sketches or signals some ideas, results, and proposals connected with the theoretical issues related to the categorial approach to language which originated from the first author (1985, 1989, 1991, 1998) and which form the basis for further research by the second author. The main aims are the following: 1) to bring into common use some Polish ideas concerned with classical categorial grammar; 2) to take into consideration a universal and simultaneously formal-logical perspective; 3) to consider Peirce's well-known differentiation of linguistic objects, i.e. their twofold ontological status as tokens (concretes) and types (abstract objects) and, according to this, to consider the biaspectual formalization of language dealing with the two main orientations in the controversy between nominalism and Platonism; 4) to characterize language according to Frege's ontological canons, according to which each expression of language corresponds to its denotation. All of these ...
Determination of administering knowledge by a rational agent acting in compliance with certain pr... more Determination of administering knowledge by a rational agent acting in compliance with certain protocols within a real system of interaction, a system of communicating, requires postulating rationality of acting by the agent, as well as postulating restriction of this rationality appropriately to the real actions.
Z badań nad Teorią Zdań Odrzuconych; Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu , 1969
This is the PhD dissertation, written under supervision of Professor Jerzy Słupecki, published in... more This is the PhD dissertation, written under supervision of Professor Jerzy Słupecki, published in the book: U.Wybraniec-Skardowska i Grzegorz Bryll "Z badań nad teorią zdań odrzuconych" ( "Studies of theory of rejected sentences"), Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Seria B: Studia i Monografie nr 22, pp. 5-131. It is the first, original publication on the theory of rejected sentences on which are based, among other, papers: "Theory of rejected propositions. I"and "Theory of rejected propositions II" with Jerzy Słupecki and Grzegorz Bryll
One well known problem regarding quantifiers, in particular the 1st-order quantifiers, is connect... more One well known problem regarding quantifiers, in particular the 1st-order quantifiers, is connected with their syntactic categories and denotations. The unsatisfactory efforts to establish the syntactic and ontological categories of quantifiers in formalized first-order languages can be solved by means of the so called principle of categorial compatibility formulated by Roman Suszko, referring to some innovative ideas of Gottlob Frege and visible in syntactic and semantic compatibility of language expressions. In the paper the principle is introduced for categorial languages generated by the Ajdukiewicz’s classical categorial grammar. The 1st-order quantifiers are typically ambiguous. Every 1st-order quantifier of the type k > 0 is treated as a two-argument functor-function defined on the variable standing at this quantifier and its scope (the sentential function with exactly k free variables, including the variable bound by this quantifier); a binary function defined on denotati...
Logical and philosophical literature provides different classifications of reasoning. In the Poli... more Logical and philosophical literature provides different classifications of reasoning. In the Polish literature on the subject, for instance, there are three popular ones accepted by representatives of the Lvov-Warsaw School: Jan Łukasiewicz, Tadeusz Czeżowski and Kazimierz Ajdukiewicz (Ajdukiewicz in Logika pragmatyczna [Pragmatic Logic]. PWN, Warsaw (1965, 2nd ed. 1974). Translated as: Pragmatic Logic. Reidel & PWN, Dordrecht, 1975). The author of this paper, having modified those classifications, distinguished the following types of reasoning: (1) deductive and (2) non-deductive, and additionally two types of them in each of the two, depending on the manner of combining their premises with the conclusion through the relation of classical logical entailment. Consequently, the four types of reasoning: unilateral deductive (incl. its sub-types: deductive inference and proof), bilateral deductive (incl. complete induction), and reductive (incl. the sub-types: explanation and verificat...
The paper is an attempt at a logical explication of some crucial notions of current general seman... more The paper is an attempt at a logical explication of some crucial notions of current general semantics and pragmatics. A general, axiomatic, formal-logical theory of meaning and interpretation is outlined in this paper. In the theory, according to the tokentype distinction of Peirce, language is formalised on two levels: first as a language of tokenobjects (understood as material, empirical, enduring through time-and space objects) and then-as a language of type-objects (understood as abstract objects, as classes of tokens). The basic concepts of the theory, i.e. the notions: meaning, denotation and interpretation of well-formed expressions (wfes) of the language are formalised on the type-level, by utilising some semantic-pragmatic primitive notions introduced on the token-level. The paper is divided into two parts. In Part I a theory of meaning and denotation is proposed, and in Part II-its expansion to the theory of meaning and interpretation is presented. The meaning a wfe is defined as an equivalence class of the relation possessing the same manner of using types (cf. Ajdukiewicz [1934], Wittgenstein [1953]). The concept of denotation is defined by means of the relation of referring which holds between wfe-types and objects of reality described by the given language.
The Warsaw School of Logic (WSL) was the famous branch of the Lvov-Warsaw School (LWS) – the most... more The Warsaw School of Logic (WSL) was the famous branch of the Lvov-Warsaw School (LWS) – the most important movement in the history of Polish philosophy. Logic made the most important field in the activities of the WSL. The aim of this work is to highlight the role and significance of the WSL in the history of logic in the 20th century. Keywords: history of logic, program assumptions and ideas of the WSL, Łukasiewicz, Leśniewski, Tarski, major achievements of representative of the WSL, importance of the WSL in the history of logic in the 20th century.
STUDIES IN LOGIC, GRAMMAR AND RHETORIC 20 (33) 2010, 2010
The considerations presented in this work are an attempt at giving an
answer to the arising doubt... more The considerations presented in this work are an attempt at giving an answer to the arising doubts: it is obvious to philosophers and logicians that such considerations must be grounded on a relevant conception of the truth and the lie, on bringing up one of the most difficult and disturbing philosophical problems, that is the problemate of the truth, on investigating what the lie is. The confusion about the notions related to the ambiguous terms of “the truth” and “the lie” introduces, in turn, a confusion connected with attempts at answering the questions posed. Thus, in the first part of this paper, we will deal with the very notion itself, or – more precisely – with the notions of the truth; in the second one– with the notions of the lie, and in the third part – we will juxtapose the notions of the truth and the lie in such a way that in each case it should be possible to provide an answer to the question asked in the title of the work. Part four, being the final one, contains certain summary of it, as well as final considerations as a peculiar challenge.
This article explores the question of how the members of the Lvov-Warsaw School promoted values t... more This article explores the question of how the members of the Lvov-Warsaw School promoted values that can be regarded as components of so-called logical culture. The author argues that these values are strictly connected with science. With references to Łukasiewicz, Czeżowski, and Kotarbiński, the article explores how values shape the logical culture and determines society as directed towards values. The article connects the meta-philosophical perspective with the philosophical one.
Lata sześćdziesiąte przyniosły z jednej strony akceptację zjawiska nieostrości w języku i nauce, ... more Lata sześćdziesiąte przyniosły z jednej strony akceptację zjawiska nieostrości w języku i nauce, z drugiej — nowe, zorientowane semantycznie lub informatycznie, rozwiązania samego problemu nieostrości. Rezultaty tych rozwiązań
okazały się jednak nie dość zadowalające. Świadczą o tym chociażby wciąż nowe
próby i poszukiwania w tym zakresie, prowadzone w różnych ośrodkach, w tym w
Polsce (Z. Muszyński (red.) 1988). Prekursorem formalnego podejścia do problemu
nieostrości i w ogóle badań nad nieostrością w Polsce jest logik — T. Kubiński (1958),
który zbudował nieklasyczny rachunek nazw nieostrych. W tym miejscu należy też
zaznaczyć, że w 1982 roku powstała formalna teoria komplementarna w stosunku do
teorii Zadeha — teoria zbiorów przybliżonych. Zbudował ją znany polski informatyk
— Z. Pawlak (1982, 1991).
Teoria Pawlaka podchodzi do zagadnienia nieostrości nienumerycznie, w przeciwieństwie do ilościowej charakterystyki zjawiska nieostrości przez Zadeha. Porusza ona aspekty jakościowe i oparta jest na idei zbioru aproksymowanego przez parę zbiorów zwanych d o l n y m i g ó r n y m p r z y b l i ż e n i e m , z b i o r u . Przybliżenia te określają, odpowiednio, pozytywny i negatywny zakres pojęcia nieostrego. Podejście Pawlaka wiąże się z odniesieniem do pojęcia wiedzy podmiotu poznającego o obiektach badanej rzeczywistości (por. Pawlak 1992). Ma więc ono jednocześnie charakter epistemiczny. Wiedza owa jest determinowana przez układ pojęć, a ten z kolei — przez system ich zakresów. Gdy zakres pojęcia jest nieostry, wyznacza go zbiór przybliżony, rozumiany jako rodzina zbiorów mających te same przybliżenia dolne i górne. Zasadnicze idee koncepcji Pawlaka miały ogromny wpływ na zbudowanie przeze mnie logicznej teorii pojęć nieostrych, której zarys jest przedstawiony w dalszej części tej pracy, przy czym inspiracji dostarczyły mi także pomysły i rozwiązania innych badaczy wymienionych w tej części prac
An attempt at reaching the past, extracting the essence of humanity, creative achievements and ma... more An attempt at reaching the past, extracting the essence of humanity, creative achievements and making them real from already partially obliterated traces is not an easy task from the personal perspective.
The Paideia Archive: Twentieth World Congress of Philosophy, 1998
This paper sketches or signals some ideas, results, and proposals connected with the theoretical ... more This paper sketches or signals some ideas, results, and proposals connected with the theoretical issues related to the categorial approach to language which originated from the first author (1985, 1989, 1991, 1998) and which form the basis for further research by the second author. The main aims are the following: 1) to bring into common use some Polish ideas concerned with classical categorial grammar; 2) to take into consideration a universal and simultaneously formal-logical perspective; 3) to consider Peirce's well-known differentiation of linguistic objects, i.e. their twofold ontological status as tokens (concretes) and types (abstract objects) and, according to this, to consider the biaspectual formalization of language dealing with the two main orientations in the controversy between nominalism and Platonism; 4) to characterize language according to Frege's ontological canons, according to which each expression of language corresponds to its denotation. All of these ...
Determination of administering knowledge by a rational agent acting in compliance with certain pr... more Determination of administering knowledge by a rational agent acting in compliance with certain protocols within a real system of interaction, a system of communicating, requires postulating rationality of acting by the agent, as well as postulating restriction of this rationality appropriately to the real actions.
Z badań nad Teorią Zdań Odrzuconych; Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu , 1969
This is the PhD dissertation, written under supervision of Professor Jerzy Słupecki, published in... more This is the PhD dissertation, written under supervision of Professor Jerzy Słupecki, published in the book: U.Wybraniec-Skardowska i Grzegorz Bryll "Z badań nad teorią zdań odrzuconych" ( "Studies of theory of rejected sentences"), Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Seria B: Studia i Monografie nr 22, pp. 5-131. It is the first, original publication on the theory of rejected sentences on which are based, among other, papers: "Theory of rejected propositions. I"and "Theory of rejected propositions II" with Jerzy Słupecki and Grzegorz Bryll
One well known problem regarding quantifiers, in particular the 1st-order quantifiers, is connect... more One well known problem regarding quantifiers, in particular the 1st-order quantifiers, is connected with their syntactic categories and denotations. The unsatisfactory efforts to establish the syntactic and ontological categories of quantifiers in formalized first-order languages can be solved by means of the so called principle of categorial compatibility formulated by Roman Suszko, referring to some innovative ideas of Gottlob Frege and visible in syntactic and semantic compatibility of language expressions. In the paper the principle is introduced for categorial languages generated by the Ajdukiewicz’s classical categorial grammar. The 1st-order quantifiers are typically ambiguous. Every 1st-order quantifier of the type k > 0 is treated as a two-argument functor-function defined on the variable standing at this quantifier and its scope (the sentential function with exactly k free variables, including the variable bound by this quantifier); a binary function defined on denotati...
The idea of rejection originated by Aristotle. The notion of rejection was introduced into formal... more The idea of rejection originated by Aristotle. The notion of rejection was introduced into formal logic by Łukasiewicz [20]. He applied it to complete syntactic characterization of deductive systems using an axiomatic method of rejection of propositions [22, 23]. The paper gives not only genesis, but also development and generalization of the notion of rejection. It also emphasizes the methodological approach to biaspectual axiomatic method of characterization of deductive systems as acceptance (asserted) systems and rejection (refutation) systems, introduced by Łukasiewicz and developed by his student Słupecki, the pioneers of the method, which becomes relevant in modern approaches to logic.
Towards Mathematical Philosoph, Series: Trends in Logic, vol.28, Springer, Berlin Heidelberg, 2009
edge with cognition of reality. A logical explication of the concept of language knowledge concei... more edge with cognition of reality. A logical explication of the concept of language knowledge conceived of as a kind of codified knowledge is taken into account in the paper. Formal considerations regarding the notions of meta-knowledge (logical knowledge about language knowledge) and truth are developed in the spirit of some ideas presented in the author’s earlier papers (1991, 1998, 2001a,b, 2007a,b,c) treating about the notions of meaning, denotation and truthfulness of well-formed expressions (wfes) of any given categorial language. Three aspects connected with knowledge codified in language are considered, including: 1) syntax and two kinds of semantics: intensional and extensional, 2) three kinds of non-standard language models and 3) three notions of truthfulness of wfes. Adequacy of language knowledge to cognitive objects is understood as an agreement of truthfulness of sentences in these three models. Keywords: Meta-knowledge, categorial syntax, meaning, denotation, categorial semantics, nonstandard models, truthfulness, language knowledge adequacy
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Keywords: history of logic, program assumptions and ideas of the WSL, Łukasiewicz, Leśniewski, Tarski, major achievements of representative of the WSL, importance of the WSL in the history of logic in the 20th century.
answer to the arising doubts: it is obvious to philosophers and logicians that such considerations must be grounded on a relevant conception of the truth and the lie, on bringing up one of the most difficult and disturbing philosophical problems, that is the problemate of the truth, on investigating what the lie is. The confusion about the notions related to the ambiguous terms of “the truth” and “the lie” introduces, in turn, a confusion connected with attempts at answering the questions posed.
Thus, in the first part of this paper, we will deal with the very notion
itself, or – more precisely – with the notions of the truth; in the second one– with the notions of the lie, and in the third part – we will juxtapose the notions of the truth and the lie in such a way that in each case it should be possible to provide an answer to the question asked in the title of the work. Part four, being the final one, contains certain summary of it, as well as final considerations as a peculiar challenge.
okazały się jednak nie dość zadowalające. Świadczą o tym chociażby wciąż nowe
próby i poszukiwania w tym zakresie, prowadzone w różnych ośrodkach, w tym w
Polsce (Z. Muszyński (red.) 1988). Prekursorem formalnego podejścia do problemu
nieostrości i w ogóle badań nad nieostrością w Polsce jest logik — T. Kubiński (1958),
który zbudował nieklasyczny rachunek nazw nieostrych. W tym miejscu należy też
zaznaczyć, że w 1982 roku powstała formalna teoria komplementarna w stosunku do
teorii Zadeha — teoria zbiorów przybliżonych. Zbudował ją znany polski informatyk
— Z. Pawlak (1982, 1991).
Teoria Pawlaka podchodzi do zagadnienia nieostrości nienumerycznie, w przeciwieństwie do ilościowej charakterystyki zjawiska nieostrości przez Zadeha. Porusza ona aspekty jakościowe i oparta jest na idei zbioru aproksymowanego przez parę zbiorów zwanych d o l n y m i g ó r n y m p r z y b l i ż e n i e m , z b i o r u . Przybliżenia te określają, odpowiednio, pozytywny i negatywny zakres pojęcia nieostrego. Podejście Pawlaka wiąże się z odniesieniem do pojęcia wiedzy podmiotu poznającego o obiektach badanej rzeczywistości (por. Pawlak 1992). Ma więc ono jednocześnie charakter epistemiczny. Wiedza owa jest determinowana przez układ pojęć, a ten z kolei — przez system ich zakresów. Gdy zakres pojęcia jest nieostry, wyznacza go zbiór przybliżony, rozumiany jako rodzina zbiorów mających te same przybliżenia dolne i górne. Zasadnicze idee koncepcji Pawlaka miały ogromny wpływ na zbudowanie przeze mnie logicznej teorii pojęć nieostrych, której zarys jest przedstawiony w dalszej części tej pracy, przy czym inspiracji dostarczyły mi także pomysły i rozwiązania innych badaczy wymienionych w tej części prac
Keywords: history of logic, program assumptions and ideas of the WSL, Łukasiewicz, Leśniewski, Tarski, major achievements of representative of the WSL, importance of the WSL in the history of logic in the 20th century.
answer to the arising doubts: it is obvious to philosophers and logicians that such considerations must be grounded on a relevant conception of the truth and the lie, on bringing up one of the most difficult and disturbing philosophical problems, that is the problemate of the truth, on investigating what the lie is. The confusion about the notions related to the ambiguous terms of “the truth” and “the lie” introduces, in turn, a confusion connected with attempts at answering the questions posed.
Thus, in the first part of this paper, we will deal with the very notion
itself, or – more precisely – with the notions of the truth; in the second one– with the notions of the lie, and in the third part – we will juxtapose the notions of the truth and the lie in such a way that in each case it should be possible to provide an answer to the question asked in the title of the work. Part four, being the final one, contains certain summary of it, as well as final considerations as a peculiar challenge.
okazały się jednak nie dość zadowalające. Świadczą o tym chociażby wciąż nowe
próby i poszukiwania w tym zakresie, prowadzone w różnych ośrodkach, w tym w
Polsce (Z. Muszyński (red.) 1988). Prekursorem formalnego podejścia do problemu
nieostrości i w ogóle badań nad nieostrością w Polsce jest logik — T. Kubiński (1958),
który zbudował nieklasyczny rachunek nazw nieostrych. W tym miejscu należy też
zaznaczyć, że w 1982 roku powstała formalna teoria komplementarna w stosunku do
teorii Zadeha — teoria zbiorów przybliżonych. Zbudował ją znany polski informatyk
— Z. Pawlak (1982, 1991).
Teoria Pawlaka podchodzi do zagadnienia nieostrości nienumerycznie, w przeciwieństwie do ilościowej charakterystyki zjawiska nieostrości przez Zadeha. Porusza ona aspekty jakościowe i oparta jest na idei zbioru aproksymowanego przez parę zbiorów zwanych d o l n y m i g ó r n y m p r z y b l i ż e n i e m , z b i o r u . Przybliżenia te określają, odpowiednio, pozytywny i negatywny zakres pojęcia nieostrego. Podejście Pawlaka wiąże się z odniesieniem do pojęcia wiedzy podmiotu poznającego o obiektach badanej rzeczywistości (por. Pawlak 1992). Ma więc ono jednocześnie charakter epistemiczny. Wiedza owa jest determinowana przez układ pojęć, a ten z kolei — przez system ich zakresów. Gdy zakres pojęcia jest nieostry, wyznacza go zbiór przybliżony, rozumiany jako rodzina zbiorów mających te same przybliżenia dolne i górne. Zasadnicze idee koncepcji Pawlaka miały ogromny wpływ na zbudowanie przeze mnie logicznej teorii pojęć nieostrych, której zarys jest przedstawiony w dalszej części tej pracy, przy czym inspiracji dostarczyły mi także pomysły i rozwiązania innych badaczy wymienionych w tej części prac
Keywords: Meta-knowledge, categorial syntax, meaning, denotation, categorial semantics, nonstandard models, truthfulness, language knowledge adequacy