Search

This paper discusses the theoretical assumptions behind the conception of the logic of faith and deed (LF&D) and outlines its formal-axiomatic frame and its method of construction, which enable us to understand it as a kind of deductive science. The paper is divided into several sections, starting with the logical analysis of the ambiguous terms of ‚faith’ and ‚action’, and focusing in particular on the concepts of religious faith and deed as a type of conscious activity relating to a matter or matters of social importance. After outlining the main ideas and basic assumptions of the theoretical conception of the LF&D as an axiomatic theory, the author introduces some axiom systems for: 1) the logics of faith LF (doxastic logics), 2) the logic of deed LD, and 3) certain logics of norms DL (deontic logics) connected with „duties” and concerning actions/deeds. Lastly, the paper outlines the scientific LF&D based on the three types of logic 1)–3).

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.

The subject matter of this work covers the issues or problems listed below: O The problem of the ontological status of language signs and a more general philosophical problem connected with it: O 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 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.

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 (CPC). We illustrate them by means of many squares of opposition and, corresponding to them—octagons, hexagons or other geometrical objects.

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 general one being: the compatibility of the 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 of syntax 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 ex- tensional 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 wellformed 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 seman- tics 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.

The idea of rejection originated by Aristotle. The notion of rejection was introduced into formal logic by Łukasiewicz. He applied it to complete syntactic characterization of deductive systems using an axiomatic method of rejection of propositions. 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 Slupecki, the pioneers of the method, which becomes relevant in modern approaches to logic.

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 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.

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 (among others, Ajdukiewicz, Leśniewski, Łukasiewicz, Tarski), 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.

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.

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 (language as an expression of thought, or thought as an expression of language), 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 formalisms. Tomention just one key instance, the notion of time has always been present in grammars (whatever their normative or descriptive stance or theoretical affiliation), whereas logical frameworks either lack explicit reference to time, or treat only some selected features, falling far short of all the features of time and aspect that natural languages are capable of expressing. Even so, the goal of translating utterances to logical forms has been a never-ending source of attraction for linguists aiming for methodological rigor and metalinguistic precision. Naturally, simple conversion of utterances to some logical form does not exhaust all their

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 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.

The approach to rough set theory proposed in this paper is based on the mutual correspondence of the concepts of extension and intension. It is different from the well-known approaches in the literature in that the upper approximations and the lower approximations of ‘unknown’ sets are considered as certain families of ‘known’ sets. This approach makes it possible to formulate necessary and sufficient conditions for the existence of operations on rough sets, which are analogous to classical operations on sets. The basic results presented in this paper, based on certain ideas of the second author, were formulated by the first author in his doctoral dissertation prepared under the supervision of the third author.

The palier explains the methods of approximation of set which were earlier put forth by the authors (1995a, 1995b) and by W. Marek and H. Rasiowa (1986). An essential part of this paper is devoted to the presentation of a certain calculus of rough sets and contextual rough sets in contextual spaces.

In this chapter, a rough description logic is built on the basis of a pragmatic standpoint of representation of knowledge. The pragmatic standpoint has influenced the acceptance of a broader definition of the semantic network than that appearing in the literature. The definition of the semantic network is a motivation of the introduced semantics of the language of the descriptive logic. First, the theoretical framework of representation of knowledge that was proposed in the papers [24,25] is adjusted to the description of data processing. The pragmatic system of knowledge representation is determined, as well as situations of semantic adequacy and semantic inadequacy for represented knowledge are defined. Then, it is shown that general information systems (generalized information systems in Pawlak’s sense) presented in the paper [5] can be interpreted in pragmatic systems of knowledge representation. Rough sets in the set-theoretical framework proposed in papers [7,8] are defined for the general information systems. The pragmatic standpoint about objects is also a motivation to determine a model of semantic network. This model is considered as a general information system. It determines a formal language of the descriptive logic. The set-theoretical framework of rough sets, which was introduced for general information systems, makes it possible to describe the interpretation of this language in the theory of rough sets. Therefore this interpretation includes situations of semantic inadequacy. At the same time, for the class of all interpretations of this type, there exists a certain descriptive logic, which — in this chapter — is called rough pragmatic description logic.

The paper proposes a new formal approach to vagueness and vague sets taking inspirations from Pawlak's rough set theory. Following a brief introduction to the problem of vagueness, an approach to conceptualization and representation of vague knowledge is presented from a number of different perspectives: those of logic, set theory, algebra, and computer science. The central notion of the vague set, in relation to the rough set, is defined as a family of sets approximated by the so called lower and upper limits. The family is simultaneously considered as a family of all denotations of sharp terms representing a suitable vague term, from the agent's point of view. Some algebraic operations on vague sets and their properties are defined. Some important conditions concerning the membership relation for vague sets, in connection to Blizard's multisets and Zadeh's fuzzy sets, are established as well. A classical outlook on a logic of vague sentences (vague logic) based on vague sets is also discussed.

The subject-matter of the consideration touches the problem of vagueness. The notion of the rough set, originated by Zdzislaw Pawlak, was constructed under the influence of vague information and methods of shaping systems of notions leading to conceptualization and representation of vague knowledge, so also systems of their scopes as some vague sets. This paper outlines some direction of searching for a solution to this problem. In the paper, in connection to the notion of the rough set, the notion of a vague set is introduced. Some operations on these sets and their properties are discussed. The considerations intend to take into account a classical approach to reasoning, based on vague premises, and suggest finding a logic of vague sentences as a non-classical logic in which all counterparts of tautologies of classical logic are laws.

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 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.

This paper presents a generalization of Pawlak’s conception of rough sets [6] and [7]. It is more general than Pawlak’s solution of the problem of the definability of sets, the knowledge of which is incomplete and vague. The authors’ conception is based on conception of contextual space [4], which was inspired by Ziarko’s approach [12] to rough sets. Rough sets introduced by Pawlak [6] are particular cases of contextual rough sets defined in the contextual approximation space. This space is defined axiomatically by means of so called context relations. Every contextual rough set determined by set X can be determined by the union of the lower approximation of X and a subset of the boundary of X. One of the important notions of the conception is the notion of an element of a contextual rough set which allows for formulating and proving the counterpart of the axiom of extensionality for contextual rough sets.

Discovery of elementary knowledge and its constituents, i.e. information contained in objects of reality is realized through asking questions including certain aspects called attributes in this paper. We describe a fragment of a discovered reality as an information system (cf. Pawlak [1,3,4]), which consists of the universum U of all the objects of this reality we are concerned with, and of a set A of attributes understood as functions each of which assigns to every object of U 1) a value of given attribute belonging to A or 2) an interval of approximate values of this attribute, i.e. an established set of possible values of this attribute. From the point of view of the cognitive agent and his knowledge of the attributes of this information system, such system is a complete system when the values of all its attributes for any objects are known, contrary to incomplete system, in which value or, respectively, set of values for an attribute are unknown or are not available, though at the same time the only possible. A complete system in the case of 1) is called by us exact and in the case of 2) is called by us approximate. An incomplete system is called by us inexact, or uncertain, or vague, in the case of 1) or strongly uncertain, or strongly vague in the case of 2).

In this work results of two experiments on application of rough set theory in land improvement sciences are included.

The discussions of the theory TSCL (B) of simple categorial languages, which is to be constructed in this chapter, will be concerned with an arbitrary but fixed language L of expression tokens which do not include variables bound by operators. The Language L, called here a simple categorial language of expression tokens, is determined by certain conditions imposed on its syntax in accordance with the idea of K.Ajdukiewicz’s categorial grammar 119351 and the conceptions of those researchers who referred to his ideas. (4)

The theory TSCL described languages which are poor in a certain sense. This is because most theorems formulated in mathematical theories are formulated in languages which include new expressions, not occurring in simple categorial languages, namely expressions with variables and operators that bind those variables. It seems, therefore, natural to modify TSCL - while retaining the main objectives described in Sec.IIA and observed when TSCL was constructed - so that it should cover the languages mentioned above. The syntactic analysis of expressions, correct from the point of view of the theory of syntactic category, should be respected, too.

Let us imagine the situation of a user of an ethnic language who wants to write a letter or the situation of a mathematician who wants to formulate theorems or to produce proofs in a formalized language. Both of them select ahd write out simple labels of’ a definite shape, combine them into more complex labels that follow one another, and -frequently reproduce them. Linguistic expressions used by a given language user are concrete, physical, sensorily perceivable objects, namely label tokens (in Latin (labels) in concreto, in English also label events; see Ch.S.Peirce [1931–19351; R.Carnap 119427). A precise explanation of the properties of those material objects encounters greater difficulties than those which we face when we establish the properties of ideal, abstract objects. We face such difficulties when we want to formulate an intuitive characteristic of a given language. (1)

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].