Movatterモバイル変換

This is a collection of new investigations and discoveries on the history of a great tradition, the Lvov-Warsaw School of logic , philosophy and mathematics, by the best specialists from all over the world. The papers range from historical considerations to new philosophical, logical and mathematical developments of this impressive School, including applications to Computer Science, Mathematics, Metalogic, Scientific and Analytic Philosophy, Theory of Models and Linguistics.

Direct download(3 more)   Export citation   Bookmark   7 citations  

This book presents a formal and philosophical analysis of language syntax. It refers to some ideas of E.Husserl and G. Frege, to S. Leśniewski's theory of syntactic categories and K. Ajdukiewicz's conception of formal grammar, also to Ch.S. Pierces's distinction between tokens (concrete linguistic entities) and types (ideal linguistic entities) and to A.A. Markov's theory of algorithms. The central aim of the book is - in the spirit of these ideas - to provide both strict yet comprehensive lectures on two (...) axiomatic theories of languages (grammars) irrespective of specific structure of their expression and the notation used in them. The main feature of these theories are that definitions of well-formed expression allow the formulation of algorithms for the examination of syntactic correctness of expressions and that their formalizations are bi-level, in reference to opposite philosophical orientations: nominalistic and idealistic. The theoretical considerations in the book speak in favour of the former. The book contains a translation of the basic contents of the book in Polish "Teorie języków syntaktycznie kategorialnych ("Theories of syntactically categorial languages"), PWN, Warszawa-Wrocław 1985, and extensive Introduction and Final Remarks. In Introduction are discussed the main assumptions, objectives and conditionings of presented theories and intuitive foundations of these theories. Final Remarks are connected with the subject-matter of the book and the ability to build syntax theories in the opposition spirit, because of the Platonic approach to language syntax. -/- . (shrink)

Direct download(2 more)   Export citation   Bookmark   17 citations  

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. (shrink)

Direct download(3 more)   Export citation   Bookmark   4 citations  

The idea of rejection of some sentences on the basis of others comes from Aristotle, as Jan Łukasiewicz states in his studies on Aristotle's syllogistic [1939, 1951], concerning rejection of the false syllogistic form and those on certain calculus of propositions. Short historical remarks on the origin and development of the notion of a rejected sentence, introduced into logic by Jan Łukasiewicz, are contained in the Introduction of this paper. This paper is to a considerable extent a summary of papers (...) which are not easily available, even to the Polish reader: (1) J. Słupecki, Funkcja Łukasiewicza (Łukasiewicz’s function), Zeszyty Naukowe Uniwersytetu Wrocławskiego, Seria B, nr 3 (1959), 33-40; (2) U. Wybraniec-Skardowska, Teoria zdań odrzuconych (Theory of Rejected Sentences), (doctoral dissertation under the supervision of Jerzy Słupecki, published as a monograph), Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Studia i Monografie, Nr 22 (1969), 5-131; (3) G. Bryll, Kilka uzupełnień teorii zdań odrzuconych (Some supplements to the theory of rejcted sentences), Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Studia i Mnografie, Nr 22 (1969), 153-154. The paper also contains a good number of results which have not been published yet. Chapter I contains results presented in the papers (1)-(3). It provides an extension of Alfred Tarski’s theory of deductive systems presented by him in the papers [1930]: Fundamentale Begriffe der Methodologie der deduktiven Wssenschaften, I and Über einige fundamentale Begriffe der Metamathematik. The enriched theory is marked with T. The most essential new concept in T is the function Cn-1, whose definition was given by Słupecki in (1) on the basis of the so-called Tarski’s general theory of deductive systems. It has the form: y e Cn-1X iff Ex e X (x e Cn {y}), where Cn is Tarski’s consequence operation. In accordance with the definition: A sentence y is rejected on the basis of the sentences of the set X iff at least one of sentences of X is a consequence of y (is deducible from y). The intuitive meaning of the rejection function Cn-1: on the basis of false sentences we can reject false sentences only (while by means of the consequence operation Cn on the basis of true sentences we can deduce true sentences only). The function Cn-1 is a generalization of the notion of rejected sentences which was introduced into logic by Łukasiewicz. The essential property of the rejection function Cn-1 is that it satisfies the axioms of general Tarski’s consequence Cn, so it is a consequence operation, called the rejection consequence. In addition, it is an additive and normal operation. In Chapter I, there are given notions analogous to those of the theory of deductive systems, but they are written down by means of the symbol ‘Cn-1’ and not ‘Cn’. There are established the properties of introduced notions and differences and analogies taking place between them and properties of respective notions of the theory T. There are also given generalizations of the notions of ‘decidable system’ and ‘consistent system’ used by Łukasiewicz. The short Chapter II contains axioms of the system T’ which is equivalent to the system T. The only difference between sets of primitive notions of these systems consists in the appearance of the function Cn-1 in the system T’ instead of the function Cn. This chapter reproduces the results given in (2), but they are partially simplified. (shrink)

Direct download(4 more)   Export citation   Bookmark   22 citations  

The paper concentrates on the problem of adequate reflection of fragments of reality via expressions of language and inter-subjective knowledge about these fragments, called here, in brief, language adequacy. This problem is formulated in several aspects, the most being: the compatibility of language syntax with its bi-level semantics: intensional and extensional. In this paper, various aspects of language adequacy find their logical explication on the ground of the formal-logical theory T of any categorial language L generated by the so-called classical (...) categorial grammar, and also on the ground of its extension to the bi-level, intensional and extensional semantic-pragmatic theory ST for L. In T, according to the token-type distinction of Ch.S. Peirce, L is characterized first as a language of well-formed expression-tokens (wfe-tokens) - material, concrete objects - and then as a language of wfe-types - abstract objects, classes of wfe-tokens. In ST the semantic-pragmatic notions of meaning and interpretation for wfe-types of L of intensional semantics and the notion of denotation of extensional semanics for wfe-types and constituents of knowledge are formalized. These notions allow formulating a postulate (an axiom of categorial adequacy) from which follow all the most important conditions of the language adequacy, including the above, and a structural one connected with three principles of compositionality. (shrink)

Direct download(5 more)   Export citation   Bookmark   7 citations  

The Introduction outlines, in a concise way, the history of the Lvov-Warsaw School—a most unique Polish school of worldwide renown, which pioneered trends combining philosophy, logic, mathematics and language. The author accepts that the beginnings of the School fall on the year 1895, when its founder Kazimierz Twardowski, a disciple of Franz Brentano, came to Lvov on his mission to organize a scientific circle. Soon, among the characteristic features of the School was its serious approach towards philosophical studies and teaching (...) of philosophy, dealing with philosophy and propagation of it as an intellectual and moral mission, passion for clarity and precision, as well as exchange of thoughts, and cooperation with representatives of other disciplines. The genesis is followed by a chronological presentation of the development of the School in the successive years. The author mentions all the key representatives of the School, accompanying the names with short descriptions of their achievements. The development of the School after Poland’s regaining independence in 1918 meant part of the members moving from Lvov to Warsaw, thus providing the other segment to the name—Warsaw School of Logic. The author dwells longer on the activity of the School during the Interwar period—the time of its greatest prosperity, which ended along with the outbreak of World War 2. Attempts made after the War to recreate the spirit of the School are also outlined and the names of continuators are listed accordingly. The presentation ends with some concluding remarks on the contribution of the School to contemporary developments in the fields of philosophy, mathematical logic or computer science in Poland. (shrink)

Direct download(3 more)   Export citation   Bookmark   2 citations  

One well known problem regarding quantifiers, in particular the 1storder 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 functorfunction 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 denotations of its two arguments is its denotation. Denotations of sentential functions, and hence also quantifiers, are defined separately in Fregean and in situational semantics. They belong to the ontological categories that correspond to the syntactic categories of these sentential functions and the considered quantifiers. The main result of the paper is a solution of the problem of categories of the 1st-order quantifiers based on the principle of categorial compatibility. (shrink)

Direct download(3 more)   Export citation   Bookmark   3 citations  

Justification of theorems plays a vital role in any rational human activity. It is indispensable in science. The deductive method of justifying theorems is used in all sciences and it is the only method of justifying theorems in deductive disciplines. It is based on the notion of proof, thus it is a method of proving theorems. In the Warsaw School of Logic (WSL) – the famous branch of the Lvov-Warsaw School (LWS) – two types of the method: axiomatic deduction method (...) and natural deduction method were developed and practiced. In this paper, both of these methods are briefly discussed with an emphasis on their historical, groundbreaking significance for logic. The axiomatic method by means of rejection (proposed by Jan Łukasiewicz – a co-creator of the WSL), which is the method of the so-called rejection proof (rejection/refutation method) in logical systems and the proving method of generalized natural deduction, which is a hybrid deduction–refutation method of proving theorems, are also outlined in the paper. The author discusses their historical significance. This paper also contains a brief mention of the most significant results which the application of the discussed methods introduced into contemporary scientific research, not only logical one. (shrink)

Direct download(3 more)   Export citation   Bookmark  

This paper reminds, puts in order, sketches and also initiates some researches from the field of logic and philosophy of language. It lays emphasis on the logical-linguistic and ontological developmental lines originated with Polish researchers. The author discusses two opposite orientations of the former line in the process of formalization of language, called here nominalistic and Platonistic. The paper mentions the author's result (1989; 1991) concerning theoretical equivalence of two axiomatic approaches to language syntax which take into consideration these two (...) different philosophical orientations, respectively. It also introduces certain formal-logical foundations of language semantics, discusses and develops the problem of the syntactic and semantic categorial agreement, i.e. the agreement of the syntactic category of each language expression with the ontological category assigned to the reference of this expression. The author, in particular, gives a solution to this problem for quantifiers. (shrink)

Direct download(5 more)   Export citation   Bookmark   13 citations  

In the paper, the authors discuss two kinds of consequence operations characterized axiomatically. The first one are consequence operations of the type Cn + that, in the intuitive sense, are infallible operations, always leading from accepted (true) sentences of a deductive system to accepted (true) sentences of the deductive system (see Tarski in Monatshefte für Mathematik und Physik 37:361–404, 1930, Comptes Rendus des Séances De la Société des Sciences et des Lettres de Varsovie 23:22–29, 1930; Pogorzelski and Słupecki in Stud (...) Logic 9:163–176, 1960, Stud Logic 10:77–95, 1960). The second kind are dual consequence operations of the type Cn − that can be regarded as anti-infallible operations leading from non-accepted (rejected, false) sentences of a deductive system to non-accepted (rejected, false) sentences of the system (see Słupecki in Funkcja Łukasiewicza, 33–40, 1959; Wybraniec-Skardowska in Teoria zdań odrzuconych, 5–131, Zeszyty Naukowe Wyższej Szkoły Inżynierskiej w Opolu, Seria Matematyka 4(81):35–61, 1983, Ann Pure Appl Logic 127:243–266, 2004, in On the notion and function of rejected propositions, 179–202, 2005). The operations of the types Cn + and Cn − can be ordinary finitistic consequence operations or unit consequence operations. A deductive system can be characterized in two ways by the following triple: $$\begin{array}{ll}{\rm by\,the\,triple}:\hspace{1.4cm} (+ , -)\hspace{0,6cm} \\ {\rm or\,by\,the\,triple}:\hspace{1.0cm} (-, +)\hspace{0,6cm}.\end{array}$$ We compare axiom systems for operations of the types Cn + and Cn −, give some methodological properties of deductive systems defined by means of these operations (e.g. consistency, completeness, decidability in Łukasiewicz’s sense), as well as formulate different metatheorems concerning them. (shrink)

Direct download(5 more)   Export citation   Bookmark   8 citations  

This paper is a continuation of Part I under the same title. Its Chapter III contains results given in the following publications: U. Wybraniec-Skardowska, Teoria zdań odrzuconych (Theory of Rejected Sentences), (doctoral dissertation under the supervision of Jerzy Słupecki, published as a monograph), Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Studia i Monografie, Nr 22 (1969), 5-131. G. Bryll, Związki logiczne pomiędzy zdaniami nauk empirycznych (Logical relations between sentences of empirical sciences). Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Studia i (...) Monografie, Nr 22 (1969), 155-216. This chapter contains an attempt to formalize some questions of methodology of empirical sciences. The theory T is enriched here not only by new definitions but also by a new primitive term and new axioms. The most important notion defined in this chapter is the set IndX of all inductive conclusions obtained on the basis of sentences of the set X. The function CnI defined by the equality: CnI X = X u IndX, satisfies the axioms of the general theory of deductive systems. The set CnI X can be understood as a system of empirical sciences in a narrower sense. This paper is a development of the considerations of G. Bryll, included in the above-listed article by this author. (shrink)

Direct download(4 more)   Export citation   Bookmark   11 citations  

In this paper, two axiomatic theories T− and T′ are constructed, which are dual to Tarski’s theory T+ (1930) of deductive systems based on classical propositional calculus. While in Tarski’s theory T+ the primitive notion is the classical consequence function (entailment) Cn+, in the dual theory T− it is replaced by the notion of Słupecki’s rejection consequence Cn− and in the dual theory T′ it is replaced by the notion of the family Incons of inconsistent sets. The author has proved (...) that the theories T+, T−, and T′ are equivalent. (shrink)

Direct download   Export citation   Bookmark   3 citations  

In the paper, original formal-logical conception of syntactic and semantic: intensional and extensional senses of expressions of any language L is outlined. Syntax and bi-level intensional and extensional semantics of language L are characterized categorically: in the spirit of some Husserl’s ideas of pure grammar, Leśniewski-Ajukiewicz’s theory syntactic/semantic categories and in accordance with Frege’s ontological canons, Bocheński’s famous motto—syntax mirrors ontology and some ideas of Suszko: language should be a linguistic scheme of ontological reality and simultaneously a tool of its (...) cognition. In the logical conception of language L, its expressions should satisfy some general conditions of language adequacy. The adequacy ensures their unambiguous syntactic and semantic senses and mutual, syntactic, and semantic compatibility, correspondence guaranteed by the acceptance of a postulate of categorial compatibility syntactic and semantic categories of expressions of L. From this postulate, three principles of compositionality follow: one syntactic and two semantic already known to Frege. They are treated as conditions of homomorphism partial algebra of L into algebraic models of L: syntactic, intensional, and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, of course, an idealization, but only expressions with high degrees of precision of their senses, after due justification, may become theorems of science. (shrink)

Direct download(4 more)   Export citation   Bookmark   3 citations  

With reference to Polish logico-philosophical tradition two formal theories of language syntax have been sketched and then compared with each other. The first theory is based on the assumption that the basic linguistic stratum is constituted by object-tokens (concrete objects perceived through the senses) and that the types of such objects (ideal objects) are derivative constructs. The other is founded on an opposite philosophical orientation. The two theories are equivalent. The main conclusion is that in syntactic researches it is redundant (...) to postulate the existence of abstract linguistic entities. Earlier, in a slightly different form, the idea was presented in [27] and signalled in [26] and [25]. (shrink)

Direct download(4 more)   Export citation   Bookmark   10 citations  

The book is a collection of papers and aims to unify the questions of syntax and semantics of language, which are included in logic, philosophy and ontology of language. The leading motif of the presented selection of works is the differentiation between linguistic tokens (material, concrete objects) and linguistic types (ideal, abstract objects) following two philosophical trends: nominalism (concretism) and Platonizing version of realism. The opening article under the title “The Dual Ontological Nature of Language Signs and the Problem of (...) Their Mutual Relations” provides a broad introduction into the problem area connected with this differentiation, while the logic-formal characteristics of the distinction are framed in the work entitled “On the Type-Token Relationships” (Chapter 1). The basic part of the book deals with issues relating to syntax (Chapters 2-4) and semantics of language (Chapters 5-6), as well as pertaining to syntactic-semantic pragmatic questions (Chapters 7-13). Throughout the book, language, categorial language, is characterized syntactically as generated by classical categorial grammar (Chapter 2) and formalized on two opposing levels: as language of expression-tokens (level of tokens) and language of expression-types (level of types). The author’s considerations contained in Chapters 2 and 4 lead to the important philosophical conclusion that in formal-logical syntactic studies on language the assumption that expression-types constitute the primary language layer while expression-tokens make the secondary one, can be neglected; thus, this speaks in favour of the opposing standpoint—the concretistic one—in the ontology of language syntax. In the works “Meaning and Interpretations”, Parts I and II (Chapters 5 and 6), it is underlined, however, that such semantic concepts as: meaning, denotation and interpretation are defined on the types level, yet their formal definitions require making use of notions of the tokens level. The semantic notions introduced in the above-mentioned articles are also used in the following works of the present selection, under the titles: “Three Principles of Compositionality” and “On Metaknowledge and Truth” (Chapters 7 and 8). They formalize two principles of compositionality that are well known in the literature on the subject, deriving from Frege, i.e. those of meaning and of denotation; they are related to the syntactic principle of compositionality which was introduced by the author. All the three principles are, at the same time, three conditions of homomorphism of categorial language algebra into three kinds of non-standard models of language (one syntactic and two semantic ones: intensional and extensional), which allows introducing three definitions of truthfulness into these models. The next two works in the collection, entitled: “On Language Adequacy” and “What is the Sense in Logic and Philosophy of Language” (Chapters 9 and 10) concern adequacy of categorial language syntax along with its dual semantics: intensional and extensional, and categorial compatibility of any of its syntactic categories with two corresponding semantic categories: intensional and extensional, based on the compatibility the syntactic category of each language expression with the ontological category assigned to its denotatum. The well-known problem of categorial compatibility for first-order quantifiers finds its solution in the paper “Categories of First-Order Quantifiers” (Chapter 11). In the work “Logic and Ontology of Language” (Chapter 12), being in a sense a summary of the considerations presented in the preceding chapters of the book, language is treated as an ontological being, characterized in compliance with the logical conception of language proposed by Ajdukiewicz. Application—like throughout the book—of tools of classical logic and set theory has resulted in emergence of a general formal logical theory of syntax, semantics and pragmatics of language, which takes into account duality in the understanding of linguistic expressions as tokens (concretes) and types (abstract objects). In terms that take into account a functional approach to language itself, there comes out an ontological neutrality of logic with respect to existential assumptions relating to the ontological nature of linguistic expressions and their extra-linguistic ontological counterparts. The issues connected with applying logic while explaining the manner of using linguistic tokens and linguistic types to determine notions of language communication are raised and illustrated in the last chapter of the work, bearing the title “A Logical Conceptualization of Knowledge on the Notion of Language Communication”. (shrink)

Direct download(2 more)   Export citation   Bookmark   1 citation  

Building on our diverse research traditions in the study of reasoning, language and communication, the Polish School of Argumentation integrates various disciplines and institutions across Poland in which scholars are dedicated to understanding the phenomenon of the force of argument. Our primary goal is to craft a methodological programme and establish organisational infrastructure: this is the first key step in facilitating and fostering our research movement, which joins people with a common research focus, complementary skills and an enthusiasm to work (...) together. This statement—the Manifesto—lays the foundations for the research programme of the Polish School of Argumentation. (shrink)

Direct download(2 more)   Export citation   Bookmark   5 citations  

The main objective of the paper is to provide a conceptual apparatus of a general logical theory of language communication. The aim of the paper is to outline a formal-logical theory of language in which the concepts of the phenomenon of language communication and language communication in general are defined and some conditions for their adequacy are formulated. The theory explicates the key notions of contemporary syntax, semantics, and pragmatics. The theory is formalized on two levels: token-level and type-level. As (...) such, it takes into account the dual – token and type – ontological character of linguistic entities. The basic notions of the theory: language communication, meaning and interpretation are introduced on the second, type-level of formalization, and their required prior formalization of some of the notions introduced on the first, token-level; among others, the notion of an act of communication. Owing to the theory, it is possible to address the problems of adequacy of both empirical acts of communication and of language communication in general. All the conditions of adequacy of communication discussed in the presented paper, are valid for one-way communication (sender-recipient); nevertheless, they can also apply to the reverse direction of language communication (recipient-sender). Therefore, they concern the problem of two-way understanding in language communication. (shrink)

Direct download(4 more)   Export citation   Bookmark   2 citations  

In this paper, with reference to relationships of the traditional square of opposition, we establish all the relations of the square of opposition between complex sentences built from the 16 binary and four unary propositional connectives of the classical propositional calculus. We illustrate them by means of many squares of opposition and, corresponding to them—octagons, hexagons or other geometrical objects.

Direct download(3 more)   Export citation   Bookmark   4 citations  

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. -/- -/- . (shrink)

Direct download   Export citation   Bookmark   6 citations  

This note is based on a lecture delivered at the Conference on the Scien- tic Research of the Mathematical Center of Opole, Turawa, May 10-11th, 1980. A somewhat extended version will be published in the Proceedings of the Conference. At the same time it is an abstract of a part of a planned larger paper, which will involve the theory of label-tokens. The theory is included into the author's monograph in Polish "Teorie Językow Syntaktycznie Kategorialnych", PWN, Warszawa-Wrocław 1985 and into (...) its English version: Theory of Language Syntax. Categorial Approach, Kluwer Academic Publishers, Boston-London-Dordrecht 1991. (shrink)

Direct download(2 more)   Export citation   Bookmark   4 citations  

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, accordingto the token-type distinction of Peirce, language is formalised on two levels: first as a language of token-objects (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 Ia theoryof meaningand denotation is proposed, and in Part II - its expansion to the theory of meaning and interpretation is presented.The meaninga 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. (shrink)

Direct download(5 more)   Export citation   Bookmark   3 citations  

This paper deals with Tarski's first axiomatic presentations of the syntax of deductive system. Andrzej Grzegorczyk's significant results which laid the foundations for the formalization of metalogic, are touched upon briefly. The results relate to Tarski's theory of concatenation, also called the theory of strings, and to Tarski's ideas on the formalization of metamathematics. There is a short mention of author's research in the field. The main part of the paper surveys research on the theory of deductive systems initiated by (...) Tarski, in particular research on the axiomatization of the general notion of consequence operation, axiom systems for the theories of classic consequence and for some equivalent theories, and axiom systems for the theories of nonclassic consequence. In this paper the results of Jerzy Supecki's research are taken into account, and also the author's and other people belonging to his circle of scientific research. Particular study is made of his dual characterization of deductive systems, both as systems in regard to acceptance and systems in regard to rejection . Comparison is made, therefore, with axiomatizations of the theories of rejection and dual consequence, and the theory of the usual consequence operation. (shrink)

Direct download(4 more)   Export citation   Bookmark   5 citations  

By logical foundations of language syntax ontology we understand here the construction of formalized linguistic theories based on widely conceived mathematical logic and dependent on two trends in language ontology. The formalization includes exclusively the syntactic aspect of logical analysis of language characterized categorially according to Ajdukiewicz's approach [1935, 1960]. Any categorial language L is characterized formally on two levels: on one of them it concerns the language of expression-tokens, on the other one - that of expression-types. Accepting the view (...) that expression-concretes, that is expression-tokens - thus, physical objects are fundamental language layer, whereas the secondary layer for L are expression-abstracts, that is expression-types - thus, ideal objects, it is the nominalistic, concretistic philosophical view on the ontological nature of language entities that is adhered to. Supporting the view that expression-types are the basic language layer, while expression-concretes are the secondary one, we take the opposite standpoint - platonizing one. We prove that the two dual approaches towards the two-level syntax theory of language are logically eqivalent. In the scope of language syntax, both conceptions deriving from two different existential assumption are equivalent. This statement is philosophical significance, since it proves that in syntactical studies on language the assumption of the existence of abstract language beings (interpreted as classes equiform tokens) can be neglected. The idea was signalized by author in ealier paper "On the type-token relationships'' [1986) and than, in a slighly diffrent form in English "On the eliminatibility of ideal linguistic entities, Studia Logica 48,no 4 (1989), pp. 587-615. (shrink)

Direct download   Export citation   Bookmark   3 citations  

In the paper, certain rational postulates for protocols describing real communicating are introduced.These rational postulates, on the one hand, allow assigning a certain typology of real systems of interactions, which is consistent with the reality of epistemic argumentation in systems of communicating, and on the other one – defining rules of using argumentation in real situations. Moreover, the presented postulates for protocols characterize information networks and administering knowledge in real interactivity systems. Due to the epistemic character of the considerations, the (...) problem undertaken in the paper concerns working out fundamental assumptions that refer to building of epistemic logics. They allow establishing the correctness of the discourse defined by rational postulates of protocols of real communication. In the context of the presented problem there are the following two research questions distinguished: 1) How do we determine the rule of building of real dynamic epistemic logics? and 2) How should we define semantics for these logics? Within the framework of considerations relating to the research questions asked, certain epistemic operators, relativized to types of communicating, are introduced. Basic logical relations between using these operators are established for these operators. The relations are presented by a diagram called the square of epis temic operators. On the basis of these logical relations some axioms for real dynamic epistemic logics are presented. The semantics of real dynamic epistem ic logics is extended by the methods of lower and upper approximation of formula evaluating. This allows defining ‘approximation Kripke models’. The results of conceptualization of knowledge on real premises of epistemic argum entation presented in this paper can be applied to rhetoric in real systems of interaction. (shrink)

Direct download   Export citation   Bookmark   2 citations  

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.

Direct download(3 more)   Export citation   Bookmark  

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 verification), logically nonvaluable (incl. inference by analogy, statistic inference), correspond to four operators of derivability. They are defined formally on the ground of Tarski’s axiomatic theory of deductive systems, by means of the consequence operation _Cn_ (Tarski in Monatshefte Math Phys 37:361–404, 1930a, C R Soc Sci Lett Vars 23:22–29, 1930b). Also, certain metalogical properties of these operators are given, as well as their relations with Tarski’s consequence operations \(Cn^+\) ( \(Cn^+ = Cn\) ) and dual consequences \(Cn^{-1}\) (Słupecki in Zeszyty Naukowe Uniwersytetu Wrocławskiego Seria B Nr 3:33–40, 1959, Słupecki et al. in Stud Log 29:76–123, 1971, Wybraniec-Skardowska, in: Wybraniec-Skardowska, Bryll (eds) Z badań nad teorią zdań odrzuconych [Studies in the Theory of Rejected Propositions], Series B, Studia i Monografie, Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Opole, 1969), and \(Cn^-\) (Wójcicki in Bull Sect Log 2(2):54–57, 1973)). (shrink)

Direct download(4 more)   Export citation   Bookmark  

In the paper, various notions of the logical semiotic sense of linguistic expressions – namely, syntactic and semantic, intensional and extensional – are considered and formalised on the basis of a formal-logical conception of any language L characterised categorially in the spirit of certain Husserl's ideas of pure grammar, Leśniewski-Ajdukiewicz's theory of syntactic/semantic categories and, in accordance with Frege's ontological canons, Bocheński's and some of Suszko's ideas of language adequacy of expressions of L. The adequacy ensures their unambiguous syntactic and (...) semantic senses and mutual, syntactic and semantic correspondence guaranteed by the acceptance of a postulate of categorial compatibility of syntactic and semantic (extensional and intensional) categories of expressions of L. This postulate defines the unification of these three logical senses. There are three principles of compositionality which follow from this postulate: one syntactic and two semantic ones already known to Frege. They are treated as conditions of homomorphism of partial algebra of L into algebraic models of L: syntactic, intensional and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, obviously, an idealisation. The syntactic and semantic unambiguity of its expressions is not, of course, a feature of natural languages, but every syntactically and semantically ambiguous expression of such languages may be treated as a schema representing all of its interpretations that are unambiguous expressions. (shrink)

Direct download(4 more)   Export citation   Bookmark  

The main goal of this paper is to outline a general formal-logical theory of language construed as a particular ontological being. The theory itself will be referred to as an ontology of language, because it is motivated by the fact that language plays a special role: it reflects ontology, and ontology reflects the world. Linguistic expressions will be regarded as having a dual ontological status: they are to be understood as either concreta – i.e. tokens, in the sense of material, (...) physical objects – or types, in the sense of classes of tokens – i.e. abstract objects. Such a duality will then be taken into account in the logical theory of syntax, semantics and pragmatics presented here. We point to the possibility of constructing the latter on two different levels, one stemming from concreta, construed as linguistic tokens of expressions, the other from their classes – namely types, conceived as abstract, ideal beings. The aim of this work is not only to outline such a theory with respect to the dual ontological nature of the expressions of language in terms that take into account a functional approach to language itself, but also to show that the logic based on it is ontologically neutral in the sense that it is abstracted from the level at which certain existential assumptions relating to the ontological nature of these linguistic expressions and their extra-linguistic ontological counterparts (objects) would have to be embraced. (shrink)

Direct download(2 more)   Export citation   Bookmark  

The subject matter of this work covers the issues or problems listed below: * The problem of the ontological status of language signs and a more general philosophical problem connected with it: * What is language as a system of signs, which – on the one hand – serves to: 1) represent our knowledge about the reality which is being recognized, and, on the other one to: 2) a. explore and better cognize or discover it, b. describe it in an (...) adequate manner, and c. enable users to make interpersonal communication? All the pragmatic functions of language require carrying out its logical-philosophical analysis, and this means both regarding its syntax and semantics. Such an analysis is not possible without determining the following: How are signs perceived and what is their ontological nature in the so-called functional approach towards logical semiotics of natural language, founded on two ways of their usage: either as signs which we use in concrete situational or situational-language contexts or as isolated signs detached from such contexts? In the first case, they are language tokens (concretes), existing material objects perceived through the senses, with a fixed temporal-spatial location, in the other one – they are non-concretes and as such (as the majority of researchers in the field of philosophy and linguistics accept) – abstract objects, language types. The type-token distinction (differentiation between abstract and concrete) has already acquired a certain status in contemporary philosophy and is of considerable importance to metaphysics and epistemology in particular. Indeed, it is most often illustrated with reference to language signs (words, expressions) as the distinction type/token of a sign, introduced into semiotics by Ch. S. Peirce. In the semiotic analysis, and also in the linguistic one, there are used both types and tokens of signs, however, often without paying due attention to when it is said about types and when about tokens. This is related to the problems which are still considered by the philosophy of language: What is the type? What is the token (specimen)? What are the mutual relations between the type and the token? Disputes concerning providing answers to these questions are related to existential issues dealing with the ontological status of these language signs and two currents within the ontology of language, remaining under the influence of two fundamental concepts formed in the debate: nominalism and realism. The author presents in brief different stances on the above-mentioned issues or problems, as well as argues that from the logical point of view: 1) working out any theoretical conception of language must take into account its bi-aspectual characteristics: as a language of expression-tokens and as a language of expression-types; 2) advocating either of the standpoints: (a) types exist independent of their tokens, or (b) it is not so, can be omitted in syntactic considerations on language; 3) mutual relations between sign-tokens differ from the mutual relations between sign-types, yet; 4) determination of the mutual relation between sign-types and sign-tokens depends on accepting either of the standpoints: (a) or (b); 5) semantic or semantic-pragmatic concepts of language, such as meaning, denotation, interpretation should be defined exclusively for types of tokens, but their definitions require certain reference to the functions which sign-tokens perform in the language (words, expressions), or to relations between them; 6) The concept of an act of language communication differs from general language communication: the first one is defined by means of sign-tokens, whereas the other one – with the use of sign-types. (shrink)

Direct download(3 more)   Export citation   Bookmark  

The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterized axiomatically in two di erent ways. We begin by recalling the classical set P of axioms of Peano’s arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: set of natural numbers and relation of (...) inequality) proposed by Witold Wilkosz, a Polish logician, philosopher and mathematician, in 1932. The axioms W are those of ordered sets without largest element, in which every non-empty set has a least element, and every set bounded from above has a greatest element. We show that P and W are equivalent and also that the systems of arithmetic based on W or on P, are categorical and consistent. There follows a set of intuitive axioms PI of integers arithmetic, modelled on P and proposed by B. Iwanuś, as well as a set of axioms WI of this arithmetic, modelled on the W axioms, PI and WI being also equivalent, categorical and consistent. We also discuss the problem of independence of sets of axioms, which were dealt with earlier. (shrink)

Direct download   Export citation   Bookmark  

The paper enriches the conceptual apparatus of the theory of meaning and denotation that was presented in Part I (Section 3). This part concentrates on the notion of interpretation, which is defined as an equivalence class of the relation possessing the same manner of interpreting types. In this part, some relations between meaning and interpretation, as well as one between denotation an interpretational denotation are established. In the theory of meaning and interpretation, the notion of language communication has been formally (...) introduced and some conditions of correctness of communication have been formulated. (shrink)

Direct download(4 more)   Export citation   Bookmark   3 citations  

The main task of this work is not to determine the bases for a moral evaluation of the lie; neither is it to describe its negative qualification. We are interested rather in the very problemate of the truth and the lie itself, considered as a juxtaposition of two of its notions: the truth and the lie, one that aims to provide a positive – as it would seem obvious – answer to the question contained in the title of the present (...) work: Does the lie contradict the truth? (shrink)

Direct download   Export citation   Bookmark  

This article summarizes the main ideas for formalizing categorial languages genrated by classical categorial grammar originated by K. Ajdukiewicz [1935,1960].This formalization is presented in detail in the author's monographs in Polish "Teorie Języków Syntaktycznie Kategorialnych" ("Theories of Syntactically Categorical Languages"), PWN, Warszawa-Wrocław 1985 and in English "Theory of Language Syntax, Categorial Approach", Kluwer Academic Publishers, Boston-London-Dordrecht 1991.

Direct download   Export citation   Bookmark   3 citations  

The paper contains an overview of the most important results presented in the monograph of the author "Teorie Językow Syntaktycznie-Kategorialnych" ("Theories of Syntactically-Categorial Languages" (in Polish), PWN, Warszawa-Wrocław 1985. In the monograph four axiomatic systems of syntactically-categorial languages are presented. The first two refer to languages of expression-tokens. The others also takes into consideration languages of expression-types. Generally, syntactically-categorial languages are languages built in accordance with principles of the theory of syntactic categories introduced by S. Leśniewski [1929,1930]; they are connected (...) with- the Ajdukiewicz’s work [1935] which was a continuation of Leśniewski’s idea and further developed and popularized in the research on categorial grammars, by Y. Bar-Hillel [1950,1953,1964]. To assign a suitable syntactic category to each word of the vocabulary is the main idea of syntactically-categorial approach to language. Compound expressions are built from the words of the vocabulary and then a suitable syntactic-category is assigned to each of them. A language built in this way should be decidable, which means that there should exist an algorithm for deciding about each expression of it, whether it is well-formed or is syntactically connected . The traditional, originating from Husserl, understanding of the syntactic category confronts some difficulties. This notion is defined by abstraction using the concept of affiliation of two expressions to the same syntactic category. (shrink)

Direct download   Export citation   Bookmark   1 citation  

This article is a translation of the paper in Polish (Alfred Tarski - człowiek, który zdefiniował prawdę) published in Ruch Filozoficzny 4 (4) (2007). It is a personal Alfred Tarski memories based on my stay in Berkeley and visit the Alfred Tarski house for the invitation of Janusz Tarski.

Direct download   Export citation   Bookmark  

This article is a characteristic of Alfred Tarski's profile, seen from a personal perspective after a long visit to Berkeley, at the invitation of Jan Tarski, in the house where Alfred Tarski lived. It takes into account the scientific achievements and research results of Tarski, as well as certain impressions of the author of these memories concerning the exotic life of this great Polish logician and mathematician of the 20th century.

Direct download   Export citation   Bookmark  

The aim of this study is to discuss in what sense one can speak about universal character of logic. The authors argue that the role of logic stands mainly in the generality of its language and its unrestricted applications to any field of knowledge and normal human life. The authors try to precise that universality of logic tends in: (a) general character of inference rules and the possibility of using those rules as a tool of justification of theorems of every (...) science, (b) principles of correct formulating of thoughts and using language, and (c) general deductive method of reasoning, i.e in the study of language syntax and semantics. The authors present two theories of language syntax and semantics as exemplifications based on the Tarski’s famous results in metalogic: (1) a formalization of universal syntax system by Andrzej Grzegorczyk based on a new primitive notion of quotation-marks operator, and (2) an enlarged system of axiomatic theory of syntax of any categorial language of Urszula Wybraniec-Skardowska’s in which the main role plays the relation of concatenation understood as a set-theoretical function. The study concludes in arguing that the importance of the result (2) lies not only in solving some philosophical problems but also in different fields of knowledge and spheres of human life as ex. in discussion and education as well. (shrink)

Direct download(3 more)   Export citation   Bookmark  

The present essay deals with certain questions in the feld of humanistic philosophy, ethics and axiology, discussed in the light of still newer and newer challenges of our changing times. It highlights the signicant role of Professor Andrzej Grzegorczyk in solving and overcoming problems encountered in the life of man, which is based on his natural logic and incessant eorts aimed at preservation of fundamental moral values, as well as at shaping the principles of the individual and social life. The (...) views held by An- drzej Grzegorczyk, which are outlined in the work, form a certain rationalistic vision of the world and mankind. (shrink)

Direct download   Export citation   Bookmark  

The two-fold ontological character of linguistic objects revealed due to the distinction between “type” and “token” introduced by Ch. S. Peirce can be a base of the two-fold, both theoretical and axiomatic, approach to the language. Referring to some ideas included in A. A. Markov’s work [1954] (in Russian) on Theory of Algorithms and in some earlier papers of the author, the problem of formalization of the concrete and abstract words theories raised by J. Słupecki was solved. The construction of (...) the theories presented here has two levels. The axiomatic theory of label-tokens: material, physical linguistic objects, constitutes the first one. Label-types, according to the literature of the subject, are defined on the other level as equivalence classes of equiform label-tokens. Assuming the opposite point of view, one can accept that theory of label-types: abstract labels, formalized on the first level, in which it is possible to define the notion of label-token as well as the derivative notions on the second level, should become the basis of formalization of the theory of linguistic expressions and the theory of language in general. The axioms and definitions of both theories of labels: T k and T p representing the other approach to the ontology of language are included in the sequel of the abstract. The foundations of the theory of labels T k in which the primary assumption as to the label-types existence is superfluous have been referred on the basis of the author's monography "Teorie Języków Syntaktycznie Kategorialnych" ( "The Theories of Syntactically Categorial Languages"), PWN, Warszawa-Wrocław 1985. The basis of the theory of labels T p which takes into account the other position has to be presented here for the first time. Some extended ideas of the paper will also be presented in author's paper "Logiczne podstawy ontologii składni Języka" ("Logical foundations of language syntax ontology), Studia Filozoficzne 6-7 (271-272), (1988), pp. 263-284. (shrink)

Direct download(2 more)   Export citation   Bookmark   2 citations  

In the article the problem of imprecise information and concepts is considered. The theory of rough sets and the theory of fuzzy sets are used to provide an original solution to this problem.

Direct download   Export citation   Bookmark  

The work broadens – to a considerable extent – Z. Pawlak’s original method (1982, 1992) of approximation of sets. The approximation of sets included in a universum U goes on in the contextual approximation space CAS which consists of: 1) a sequence of Pawlak’s approximation spaces (U,Ci), where indexes i from set I are linearly ordered degrees of contexts (I, <), and Ci is the universum partition U, 2) a sequence of binary relations on sets included in U, relations called (...) context relations indexed with degrees of contexts. The introduction of relations of contexts of sets was inspired by the work of W. Ziarnko (1993). Intuitively, set X is the context of set Y to the degree i, when on the basis of knowledge of the elements of set X we can obtain knowledge about the belonging of these elements to set Y to the degree i which is a degree of uncertainty or ambiguity, or inaccuracy. The operations of approximation of sets Ci-, BNi, Ci+, called lower approximation of the degree i, boundary of the degree i, upper approximation of the degree i, are formulated in a way which is analogous with the systems of Pawlak’s approximation, by expanding their formulas accordingly. In the next part of the work, it is shown (in an analogous way to Bryniarski’s work (1989)) that one can define the following operations on contextual rough sets: the union of contextual rough sets, the intersection of contextual rough sets and the complement of contextual rough set. It is proved that the algebra of contextual rough sets is a ditributive lattice with a unit and with a zero. On the basis of the results presented above the problem of defining operations on contextual rough sets by membership relations (Bryniarski 1989) may be formulated analogously to the classical operations on sets. (shrink)

Direct download(3 more)   Export citation   Bookmark  

The European Summer Schools in Logic, Language and Information (ESSLLI) have been organised every year since 1989 under the auspices of the Association for Logic, Language and Information (FoLLI) in different cities around Europe. The 24th European Summer School in Logic, Language and Information (ESSLLI 2012) took place at the University of Opole, Poland, during August 6-17, 2012. The organisation committee was chaired by Janusz Czelakowski and Urszula Wybraniec-Skardowska (Institute of Mathematics and Informatics, University of Opole) and the programme committee (...) was chaired by Andreas Herzig (University of Toulouse and CNRS). During two weeks, over 340 participants from 36 countries were offered a rich academic program of their choice from 9 foundational, 16 introductory and 17 advanced courses and 6 workshops. Just as in the previous years, the 17th conference on Formal Grammar collocated with ESSLLI. Besides the regular courses, a traditional highlight of the summer school were the 4 evening lectures by distinguished academics: - Johan van Benthem (Institute for Logic, Language & Computation (ILLC), Amsterdam and Department of Philosophy, Stanford University), "Computation as Agency''; - Melvin Fitting (Department of Mathematics and Computer Science, City University of New York), "Reasoning About Games''; - Jonathan Ginzburg (UFR Études anglophones, Université Paris-Diderot (Paris 7)), "False starts, jokes, and music:semantics in the 21st century''; - Adam Przepiórkowski (Institute of Computer Science, Polish Academy of Sciences, Warsaw), "Secret life of corpora''. Another traditional event was the student session, organised by a student programme committee chaired by Rasmus K. Rendsvig (University of Copenhagen): master and doctoral students presented 16 papers and 5 posters, and Michał Zawidzki obtained the best paper award and Raul Fervari obtained the best poster award. Moreover, the Beth Dissertation Prizes were announced at the FoLLI General Meeting and went to Andreas Kapsner (University of Barcelona) and Daniel R. Licata (Carnegie Mellon University). Apart from the busy academic programme, participants of the summer school were offered an exciting social program as well. In addition to the traditional ESSLLI Party and the famous Students vs. Lecturers Soccer Match (also called `Johan van Benthem Cup'), kayaking, and excursions to Kraków and Wrocław were offered. -/- . (shrink)

Direct download(2 more)   Export citation   Bookmark  

Logic and linguistics have engaged in a many-faceted dialogue since the very beginnings of both disciplines in Antiquity. While participants may have had diverse views over the ages, arguably, the dialogue has always revolved around the relationship between human thought and natural language. While there are those who see these two domains as one and the same, or as a case of one-directional influence , we beg to differ. To us, the long historical tradition of authors such as Arnauld, Boole, (...) Turing, or Jespersen demonstrates the much richer perspectives on language and reasoning that are needed, including connections with intelligence and computability.Another major historical theme in the above dialogue are similarities or convergences between natural and artificial languages. It is clear that natural languages are not only semantically broader, but also much closer to human communication than artificial languages, and so linguistic grammars have long dealt with a larger set of problems than logical formalism. This specal issue of JoLLI consists six selected papers presented in the "Logic and Linguisics" Workshop in UNI-LOG 2013 in Rio de Janeiro, plus two papers by invited authors: Johan van Benthem, and Marcus Kracht and Uno Klein. It contains a number of samples of what is intriguing and promising work at the interfaces of logic and language. (shrink)

Direct download(3 more)   Export citation   Bookmark  

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 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 and types 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 factors make possible not only the syntactic characterization of language but also the introduction of syntactic and semantic definitions of a true expression and its denotation. These notions correspond here to the old classical, but not necessarily standard, understanding of semantic concepts. The paper is divided into four sections: the first contains a brief characterization of the categorial approach to syntax; the second presents two strains of this approach; the third touches on certain general semantic issues connected with the notion of truth; and the last gives some final remarks. (shrink)

Direct download(2 more)   Export citation   Bookmark   1 citation  

The monograph contains three works on research on the concept of a rejected sentence. This research, conducted under the supervision of Prof. Jerzy Słupecki by U. Wybraniec-Skardowska (1) "Theory of rejected sentences" and G. Bryll (2) "Some supplements of theory of rejected sentences" and (3) "Logical relations between sentences of empirical sciences" led to the construction of a theory rejected sentences and made it possible to formalize certain issues in the methodology of empirical sciences. The concept of a rejected sentence (...) was introduced by J. Łukasiewicz in the course of his inquiry into Aristotelian Syllogistic. It was essentialy generelized by J. Słupecki who, with reference to Tarski's axiomatic theory of deductive systems (1930), gave the formal definition of the rejection function Cn' according to which a sentence y is rejected on the base of a set of sentences X iff at least one sentence which belongs to X can be deduced from the sentence y. Słupecki proved (1959) that the function Cn', which assignes to the set X the set of sentences rejected on the basis of X, is additive and satisfies the axioms of Tarski's general theory of deductive systems. Work (1) uses not only Tarski's general theory of systems but also his the so-called enlarged theory of systems (1930). It constructs a theory T of rejected sentences (propositions), which is created from Tarski's richer theory of systems by adding one axiom and a number of definitions; among them the definition of the rejection function Cn' is fundamental one. Cn' is a function which assignes to every set X of false sentences the set Cn'X whose members are exclusively false senteces. A number of theorems which describes properties of defined concepts have been proved in T. Some of them are necessary for conducting considertions of the concluding section of (1) devoted to the empirical sentences. Section 2 of (1) deals with the theory of the unit consequenses. It is characterized by an axiom from which follow all axioms of Tarski's general theory of deductive systems, additivity and the property that the unit consequence of the set X of sentences is a unit consequence only one sentence of X. An example of unit consequence is the rejection consequence Cn'. In section 3 of (1) the dual theory T' equivalent to T is presented. In T' the primitive concept is the rejection function Cn'. The fundamental axiom of T' states that Cn' is a unit consequence. The concluding section brings a few examples of applications of the definitions and theorems of T in the domain of the scientific methodology. Work (2) corresonds to J. Słupecki paper (1959) and the theory T of rejected propositions given in work (1). In this paper some suplements of the theory T are given. Work (3) is an extension of the theory of rejected propositios given in the paper (1). It is an attempt of formalization some problems of methodology of the empirical sciences which concern to such concepts as: empirical sentence, similar sentence, generalization, inductive conclusion etc. (shrink)

Direct download(2 more)   Export citation   Bookmark   1 citation