STUDIES IN LOGIC, GRAMMAR AND RHETORIC 52 (65) 2017 DOI: 10.1515/slgr-2017-0054 Urszula Wybraniec-Skardowska Cardinal Stefan Wyszyński University in Warsaw, Department of Philosophy A LOGICAL CONCEPTUALIZATION OF KNOWLEDGE ON THE NOTION OF LANGUAGE COMMUNICATION* Abstract. 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. Keywords: act communication, language communication in general, token-type distinction, meaning, interpretation, problem of adequacy of communication, formal-logical theory of language communication. 1. Introduction The key issue of modern pragmatics as a part of semiotics is communication, whose main task is the transmission, processing, and transformation of information. It does not mean, however, that we fully understand what communication is and what the conditions of its proper flow are. The problem of communication is as old as mankind and has been present in many different fields ever since, for example: in cultural systems, sign systems (inISBN 978-83-7431-542-5 ISSN 0860-150X 247 Urszula Wybraniec-Skardowska cluding language systems), but also in market systems, bank systems, and recently emerged computer networks. The discovery and cognition of reality is best realized through the processes of cognition, whose result is knowledge of a conceptual space. It is expressed and represented in language and transferred to others in acts of communication by means of concrete, material language expressions – token-expressions (see Diagram 1). Diagram 1 Acts and processes of communication take place not only among people, but also among any communication channels, organization units, which are the subjects of this communication, for example: groups of people, firms, political parties, governments and so on. In communication acts, a very important role is played by the knowledge of objects represented by means of words and other signs. It can also be influenced by cultural, psychological, sociological, political, and technical factors. In this paper I concentrate on the representation of knowledge that takes place in language systems of communication. The aim of the paper is to outline a logical theory of language in which the phenomenon of language communication and language communication in general are defined and some conditions for their adequacy are formulated. Assimilation and transfer of knowledge about objects to other people is possible owing to the cognitive-communicative function of language. The transfer of verbal knowledge takes place in acts of communication by means of concrete, material language expressions (token-expressions). In formal considerations, first, we want to provide definitions of an act of language communication and the related notions such as: using linguistic tokens and interpreting linguistic tokens in order to formulate some general conditions for the correct course of the act of communication, i.e. to consider its adequacy and indicate some general causes of verbal miscommunication. 248 A Logical Conceptualization of Knowledge on the Notion of Language... The notion of an act of communication has to be differentiated from the notion of language communication which is a basic concept of logical pragmatics and of the logic of language in general. Answering the following questions: what is language communication as such? and what are the conditions for correct communication? i.e. considering the problem of its adequacy is a primary task for a general theory of language communication. A logical conceptualization of the knowledge on the notion of language communication and such related notions as meaning and interpretation of language expressions involved in communication cannot be performed unless certain philosophical assumptions concerning the nature of these notions and of the expressions themselves are adopted, and unless some prior assumptions are made on the selection of primitive notions and the method of defining. In the paper, an axiomatic theory of language communication TLC, as a semantic-pragmatic theory, independent of extra-logical factors, is outlined. First, in Section 2, some aspects that we take into account in formalization of the theory TLC will be discussed. The theory has to be based on a theory of syntax T. Some foundations of the syntax theory T will be presented in Section 3. According to the token-type distinction of language objects originated from Ch. S. Peirce (1931–1935) it is formalized on two levels: token and type. The proposed theory TLC will be developed in Section 4 as an expansion of the syntax theory T to the semanticpragmatic theory in which – on the token-level – the concept of act of language communication by means of token-expressions (understood as physical, material, empirical, enduring through time-and-space objects) will be defined, and the problem of adequacy of communicating by means of such expressions will be considered, while – on the type-level – the notion of language communication by means of type-expressions (understood as abstract, ideal objects, classes of token-expressions) and such related notions as meaning and interpretation will be defined and some conditions of adequacy for such communication will be formulated. The paper ends with Section 5, in which we differentiate the earlier given conditions for adequacy and general, logical factors for verbal miscommunication and misunderstanding from the extra-logical (e.g., psychological, sociological, political) ones. 249 Urszula Wybraniec-Skardowska 2. Three Aspects in Formalization of the TLC Theory The presentations of an axiomatic formal-logical theory TLC as a semantic-pragmatic theory, independent of any extra-logical factors, psychological or sociological or communication channels, which – on the one hand – can enhance understanding, but – on the other – can interfere with it, will be however based on some assumptions. Although TLC will concern communication by means of expressions of any language, it will take into consideration, to a certain degree, the following three aspects: (1) the cognitive-communicative function of natural language, according to its genesis, (2) the so-called functional approach to logical analysis of this language, and the one connected with it: (3) two understandings of a manner of use and a manner of interpreting language expressions in communication. Let us expand on these aspects. 2.1. The cognitive-communicative function of natural language according to its genesis Given the genesis of natural language, one can easily observe that it was formed in the process of cognition and communication between people who made use of material, concrete signs. Accordingly, we make the assumption that the primitive linguistic entities applied in communication acts between their senders and recipients are material creations, e.g. given sounds, written signs, physical objects somehow placed in time and space, concrete objects which have some referents attributed to them, and which are called tokens. According to the well-known token-type distinction made by CH.S. Peirce Diagram 2 250 A Logical Conceptualization of Knowledge on the Notion of Language... (1931–1935), we differentiate token-signs (signs-examples) from type-signs, which are abstract, ideal linguistic objects, and whose physical representations are just tokens. In acts of communication (see Diagram 2) the sender s calls, uses a token e of a sign with reference to a broadly conceived object o, while the recipient r interprets it in compliance or in discordance with the sender's intention, as object o or another object o'. Compliance produces understanding, while discordance produces misunderstanding. 2.2. The functional approach to natural language analysis As we have already seen, in order to explain the notion of communicating we had to introduce the terms using and interpreting, which entailed the use of concrete entities, i.e. tokens, and the inclusion of situational contexts accompanying them. This shows how tokens function in communication acts. Even though we are not going to refer to situational contexts in our theoretical considerations, the context is always present in such acts. In the proposed theory TLC, the basic semantic-pragmatic notions, including the notion of language communication and the related concepts – meaning and interpretation – are defined by means of expression-types, and yet their definitions involve such primitive notions of the theory as using and interpreting expression-tokens. The formal conception of language communication has some connections with the understanding of meaning as a manner of use of expressions and interpretation as a manner of interpreting expressions. Speaking about the functional approach to natural language analysis, we have to take into consideration the manner of use and the manner of interpreting language expressions. The latter will be regarded as a special case of the former. 2.3. Two understandings of manner of use and manner of interpreting language expressions in communication The functional approach to natural language analysis involves speaking about two meanings of the terms: 'a manner of use' and 'a manner of interpreting'. After the approach of J. Pelc (1971; 1979), we distinguish two understandings of these terms: • in the first of them, the manner of using (use) and the manner of interpreting (int) occur only in given circumstances, in specific languagesituational-contexts, and concern expression-tokens only, 251 Urszula Wybraniec-Skardowska • in the second – the manner of Use (usage) and the manner of interpreting (Int) characterize the meaning of the expression and the interpretation of the expression, respectively; these manners are somehow built into this meaning and this interpretation, respectively. In this case an expression can be treated as isolated, static, out of context, e.g. as an entry in a dictionary. It is then an expression-type, a class of its concrete occurrences, a distributive set of expressiontokens used either to represent a given object, or in concrete acts of communication in specific linguistic-situational contexts, with reference to only one broadly conceived object or to a set of objects of the same kind. For example, two single tokens of the expression-type 'scientist', having an established meaning (the manner of Use) or a specific interpretation (the manner of Int) in English, can be used in a similar linguistic-situational context either with reference to a given scientist, e.g. the one which I am pointing to, or with reference to two different scientists, e.g. in a situation of teaching a student the meaning of the word 'scientist' through a definition and pointing to two different scientists. The relation use and its sub-relation int, concerning all the relations of physical object-based reference of expression-tokens made by users of language, will be primitive notions of the theory TLC proposed here. The relation Use (resp. the relation Int) is, on the other hand, a relation defined by means of the relation of use (resp. the relation int) and applied by users of language for expression-types. The difference between these relations is explained by the fact that two persons can Use the same expression-type by means of its two different tokens. The notion of an expression is a syntactic one and must be defined on the basis of a theory of syntax. 3. Language Syntax; Theory T 3.1. Two levels of formalization of syntax of language The theory of syntax T is formalized on two levels: token-level and typelevel. According to the token-type distinction by Peirce (1931–1935), any language L is characterized as a construct of a double ontological nature: both as • a language of expression-tokens (at the token-level) and as • a language of expression-types (at the type-level). 252 A Logical Conceptualization of Knowledge on the Notion of Language... The theory T is first formalized on the token-level as the theory of token-syntax describing L as a language of expression-tokens, and then, on the type-level, as the theory of type-syntax describing L as a language of expression-types. The theory of type-syntax is an extension of the theory of token-syntax. Tokens are primitive objects of the theory T. They are intuitively understood as concrete, material, empirical objects, enduring through time and space and perceived by sight. They are usually inscriptions, but do not have to be inscriptions. They can be on paper, a notice board, a blackboard, a computer screen, a stone, etc.; they may be configurations of such things as jigsaw-puzzle pieces, leaves, stones, stars, or smoke signals, or illuminated advertisements, and so on. Types are derived objects of the theory T defined by means of tokens. They are understood as sets (classes) of tokens bearing an identifiability relation to each other, i.e. types are ideal, abstract entities. 3.2. Identifiability of linguistic tokens The relation of identifiability ∼ of tokens (a primitive notion of the theory T) is determined by pragmatic factors and not by physical similarity, and it is understood very broadly. For instance, inscriptions printed in different types but consisting successively of the same letters of the alphabet may be identifiable, e.g. the word-tokens: DUBROVNIK Dubrovnik Dubrovnik written in capital letters, in bold with bigger typeface or in italics, respectively, can be regarded as identifiable words. We will assume that the relation of the identifiability ∼ of tokens is an equivalence relation. The expressions of language L are defined separately on the token-level and on the type-level. They are suitable concatenations of tokens or types. The relation of concatenation of tokens is another primitive notion of the theory T. 3.3. Concatenations Concatenations of tokens are complex words of language L obtained from two words of the vocabulary of language L – the next primitive notion of theory T. Concatenations on the token-level may be, but do not have to be, sequences of two tokens. Intuitively, a concatenation of two written tokens a 253 Urszula Wybraniec-Skardowska and b, for example in a European language, is a written token c that is made up by adding the written token b∗, identifiable with b, to the token a∗, identifiable with a, on the right. For example, the concatenations of the following word-tokens: C o n FORMAL METHODS f e r e n c e the second and the first, is the name-token: Formal Methods Conference and any name-token identifiable with it, in particular the token aligned vertically: FORMAL METHODS CONFERENCE or any token written on each poster on the conference. So, the relation of concatenation defined by tokens is not a settheoretical function and the relation of identifiability is not a relation of physical similarity. These two relations and the vocabulary of tokens are primitive notions of the theory of words which is included in the theory of syntax T. They are formalized on the token-level. All of them satisfy some specific axioms of the theory. 3.3. Well-formed expressions The most important notion of the theory of syntax T is the notion of a well-formed expression of language L (for short: wfe). The theory T can be built as a theory of language syntax in which (see WybraniecSkardowska, 1991) all wfes are generated by a categorial grammar (see K. Ajdukiewicz, 1935; Y. Bar-Hillel, 1950, 1953, 1964; J. Lambek, 1958, 1961; R. Montague, 1970a, b, c, 1974; M. J. Cresswell, 1973, 1977; W. Marciszewski, 1988; W. Buszkowski, 1988, 1989 and others). On the basis of the theory T we can reconstruct such a grammar. The notion of a wfe is defined firstly on the token-level and then on the type-level. Then the set S of all wfe-tokens is formally defined as the smallest set including the vocabulary of tokens and closed with respect to syntactic connection rules. 254 A Logical Conceptualization of Knowledge on the Notion of Language... The set S of all well-formed expression-types (for short: wfe-types) is defined as the quotient family of the set S of all wfe-tokens determined by the relation ∼ of identifiability: S = S/ ∼ . Hence, we get that: p ∈ S iff ∃p ∈ S(p = [p]∼ = {q ∈ S : q ∼ p}). So, any well-formed expression-type p is an equivalence set of all wfe-tokens identifiable with a wfe-token p. In the following sections, we will use wfe-types not only as elements of the set S but also all non-empty subtypes of wfe-types of this set. By wfetypes of L we will mean all elements of the set S∗: S∗ = {e ⊆ p : e 6= ∅ ∧ p ∈ S}, i.e. all non-empty sets of identifiable wfe-tokens. 4. A Theory of Language Communication – Theory TLC 4.1. Token-level Because the formal theory TLC should define the notion of language communication, its conceptual apparatus has to refer to the notions ofmeaning and interpretation of language expressions and to empirical acts of communication among people. So, on the token-level its conceptual apparatus has to include the notions of using and interpreting token-expressions by users of language L. Thus, we accept the postulate that in communication acts the sender, in order to send the message, applies the function use connected with the object reference of a wfe-token, whereas the recipient, in order to receive the message, applies another function – the function int of interpreting tokens. 4.1.1. Primitive notions of TLC Primitive notions of the theory TLC are: – the set User of all users of a given language L, – the set Ont of all extra-linguistic objects described by L, – the two-place operation use of using the wfe-tokens of L. – the two-place operation int of interpreting the wfe-tokens of L. The first two primitive notions are understood very broadly. The set User of users of language L can be composed of current as well as former 255 Urszula Wybraniec-Skardowska or future users of this language. We do not make any assumptions, either, about the ontological nature of objects of the set Ont. They can be not only material objects, but also, for instance, fictional or abstract creations described by language L. Of course, the sets User and Ont are non-empty sets: AXIOM (sets: User, Ont): User 6= ∅ and Ont 6= ∅. We understand the operation (relation) use as an operation producing, calling, using, exposing or interpreting wfe-tokens in order to refer them to corresponding objects of the set Ont. We can also call this operation a function of object reference of wfe-tokens by users of language L. The operation int occurs when we speak about communication by means of expression-tokens. This operation will be a restriction of the former one. The operations use and int satisfy the following axioms: AXIOMS (using): use is a partial function of User× S → Ont, Dom1(use) = User and Dom2(use) ⊂ S. AXIOM (interpreting): int is a partial function of the function use, i.e. ∅ 6= int ⊆ use and Dom2(int) ⊆ Dom2(use) ⊂ S. The expression: use(u, e) = o, where u ∈ User, e ∈ S, o ∈ Ont is read: the user u uses (makes or exposes) the wfe-token e to refer to the object o. This object o is called the referent of the wfe-token e assigned by its user u. Similarly, the expression int(u, e) = o is read: the user u interprets (understands) the wfe-token e as a sign-token of the object o. The object o is called the interpretandum of the wfe-token e. It follows from the second axiom that every user of L uses at least one wfe-token of L to refer to an object. Not every wfe-tokenmust have a referent. From the third axiom it follows that the operation int of interpreting tokens is narrower than the operation use of using tokens. This is because the pair 〈a user, a token〉, which has a referent, may have no corresponding interpretandum when, for instance, this token cannot be received or was used with the intention of being interpreted by a recipient, but he/she cannot interpret it. The fact is, however, that each pair that has an interpretandum also has the same referent. The notion int of interpreting tokens emerges when we speak about communication by means of expression-tokens. From the axioms, we immediately get: 256 A Logical Conceptualization of Knowledge on the Notion of Language... COROLLARY 1 a. ∀u ∈ Dom1(int) ∀e ∈ Dom2(int) (int(u, e) = use(u, e)), b. ∃u ∈ User ∃e ∈ S ∃o ∈ Ont (use(u, e) = o = int(u, e)), c. Dom1(int) ⊆ Dom1(use) ⊆ User. Thus (see part c.), interpreting tokens is a particular case of using tokens. On the basis of part a. we can state that if we limited both domains of the operation use using wfe-tokens to the domain of operation int interpreting wfe-tokens of L, then these two operations would not be discernible; then every user using any expression-token to refer to an object is a person who also interprets this expression as this object. Such a situation is not specific of communicating by means of tokens, but it follows from part b. that there exists at least one user of L who uses and interprets a token in a given act of communication by means of this token as the same object. 4.1.2. Act of Communication The notion of communication act is new in TLC. An act of communication is defined as a triple satisfying of some conditions: DEFINITION 1a) (act of communication): 〈s, e, r〉 ∈ acom iff s, r ∈ User ∧ e ∈ S ∧ ∃o, o′ ∈ Ont (use(s, e) = o ∧ int(r, e) = o′). Its first element s (the sender) and the third of its elements r (the recipient) are users of language L, the second element e is a wfe-token of L and there exist objects o, o′ ∈ Ont such that the sender s of the expression e uses the expression e to refer to the object o (the referent) and the recipient r of the expression e interprets this expression as a sign-token of the object o′ (the interpretandum) (see Diagram 3a). Communication acts can be carried out by means of two different expression-tokens of the same wfe-type (see Diagram 3b), if the sender uses a token and the recipient interprets another token the same expression-type; this is so in e-mail, microphone or telephone communication. So, the more general definition of an act of communication is in accordance with: DEFINITION 1b) (act of communication): 〈s, e, r〉 ∈ ACom iff s, r ∈ User ∧ ∧ ∃e ∈ S∗ (e ∈ e ∧ ∃e′ ∈ e ∃o, o′ ∈ Ont (use(s, e) = o ∧ int(r, e′) = o′). 257 Urszula Wybraniec-Skardowska Diagram 3a Diagram 3b It is easy to see that any act of communication by means of one token is also an act of communication by means of two expression-tokens. So, we have COROLLARY 2. 〈s, e, r〉 ∈ acom ⇒ 〈s, e, r〉 ∈ ACom. Examples of communication acts include: making an announcement, this present paper, a specific question, e.g. in a discussion, etc. 4.1.3. Adequacy of Communication Acts The problem of adequacy of an act of communication by means of a wfetoken consists in its effectiveness. A communication act is effective if using the token by its sender and interpreting the token or a token identifiable with that token by its recipient are in agreement, i.e. the referent to which the sender uses the token and the interpretandum as an object of interpreting the token or a token identifiable with that token by its recipient, are the same. In other words, a communication act is effective when an understanding takes place between its sender and its recipient. Two definitions of an act of communication by means of wfe-tokens will bring us to two definitions of the notion of understanding (see Diagrams 4a and 4b). DEFINITION 2a) (understanding): unde(s, r) iff s, r ∈ User ∧ e ∈ S ∃o ∈ Ont (use(s, e) = o = int(r, e)). DEFINITION 2b) (understanding): Unde(s, r) iff s, r ∈ User ∧ ∃e ∈ S∗ (e ∈ e ∧ ∃e′ ∈ e ∃o ∈ Ont (use(s, e) = o = int(r, e′)). 258 A Logical Conceptualization of Knowledge on the Notion of Language... Diagram 4a Diagram 4b Abbreviations 'unde(s, r)' and 'Unde(s, r)' are used here for the expressions: 'Between s and r in an act of communication by means of the wfetoken e or the tokens: e and some identifiable token e′, respectively, there exists understanding'. The object which is both the referent and the interpretandum in the act of communication determined by Unde(s, r), is called the object of understanding. It is quite obvious that if there exists understanding in the first sense, then there exists understanding in the second sense, and the following conclusions are valid: COROLLARY 3 a) unde(s, r) ⇒ Unde(s, r), b) unde(s, r) ⇒ 〈s, e, r〉 ∈ acom, c) Unde(s, r) ⇒ 〈s, e, r〉 ∈ ACom, d) ∃u ∈ User ∃e ∈ S (〈u, e, u〉 ∈ acom ∧ unde(u, u)), e) acom 6= ∅ ∧ACom 6= ∅. Point d) of the above corollary states that there exists at least one user of the language L who takes part in an act of communication by means of an expression-token simultaneously as the sender and the recipient, and understanding takes place in the act. So, we have e): the sets of all communication acts, in both senses, are nonempty. 4.1.4. Miscommunication: Misunderstanding If, in an act of communication by means of a wfe-token, understanding does not take place between its sender and its recipient, then the act of communication is not adequate and we may speak about miscommunication. It occurs if misunderstanding takes place in this act or if an attempted act of communication fails because of non-understanding between the sender and the recipient. 259 Urszula Wybraniec-Skardowska From two definitions of a communication act, we will obtain two definitions of misunderstanding and two definitions of non-understanding. DEFINITION 3a) (misunderstanding): misunde(s, r) iff s, r ∈ User ∧ e ∈ S ∃o, o′ ∈ Ont (use(s, e) = o 6= o′ = int(r, e)). DEFINITION 3b) (misunderstanding): Misunde(s, r) iff s, r ∈ User ∧ ∧ ∃e ∈ S∗ (e ∈ e ∃e′ ∈ e ∃o, o′ ∈ Ont (use(s, e) = o 6= o′ = int(r, e′)). Diagram 5a Diagram 5b If the sender of the expression-token e uses this expression to refer to an object and the recipient interprets this or another expression-token e′ of the same type as another object, then there exists a misunderstanding between the sender and the recipient in the act of communication by means of the expression e (see Diagrams 4a and 4b). 4.1.5. Miscommunication: Non-understanding If the sender of the expression e uses e to refer to a referent but the recipient is unable to interpret the expression e or an expression identifiable with that expression, then there follows a non-understanding (see Diagrams 6a and 6b). Thus, symbolically: DEFINITION 4a). (non-understanding): non-unde(s, r) iff s, r ∈ User ∧ e ∈ S ∧ ∧ ∃o ∈ Ont (use(s, e) = o ∧ ∀o′ ∈ Ont (¬int(r, e) = o′)). 260 A Logical Conceptualization of Knowledge on the Notion of Language... DEFINITION 4b). (non-understanding): Non-unde(s, r) iff s, r ∈ User ∧ ∃e ∈ S (e ∈ e ∧ ∧ ∃e′ ∈ e ∃o ∈ Ont (use(s, e) = o ∧ ∀o′ ∈ Ont (¬int(r, e′) = o′)). Diagram 6a Diagram 6b 4.2. Type-level 4.2.1. Communication by means of expression-types Empirical communication by means of expression-tokens has to be distinguished in a given community of Users from communication by means of wfe-types. On the type-level we expand the conceptual apparatus of the TLC with new notions. The most important one is the notion of communication by means of types. It is determined as a value of an operation communication C defined on expression-types. The operation communication C is a function defined as follows: DEFINITION 5 (operation communication): C : S∗ → 2User×S×User C(e) = {〈s, e, r〉 : s, r ∈ User ∧ e ∈ e ∧ 〈s, e, r〉 ∈ ACom} for every wfe-type e of language L. The value C(e) of the function C for the expression-type e is called communication by means of the expression-type e. Communication C(e) by means of the expression-type e is the relation User×S×User consisting of all ordered triples, such that the first element (the sender) uses a wfe-token of e and the third component (the recipient) interprets a token of e in an act of communication. So, communication C(e) by means of the expressiontype e is the set of all communication acts by means of expression-tokens of the type e. 261 Urszula Wybraniec-Skardowska It includes the set of all communication acts by means of only one token of the type. Moreover, it follows from earlier corollaries that there exists a wfe-type e such that communication C(e) by means of type e is a nonempty set. Thus we arrive at: COROLLARY 4 a) {〈s, e, r〉 : s, r ∈ User∧ e ∈ e ∧ 〈s, e, r〉 ∈ acom} ⊆ C(e), b) ∃e ∈ S∗ (C(e) 6= ∅). 4.2.2. Using types and Interpreting types Users that participate in acts of communication belonging to language communication by means of an expression-type e are also Using the expression-type e: senders Use this type while recipients Interpret it. The relation Use of Using expression-types and its sub-relation Int of Interpreting expression-types are new notions of TLC. They are binary relations satisfying some axioms and defined by means of relations use and int for tokens, respectively: AXIOM (Use): Use ⊆ User× S∗, AXIOM (domain of Int): Dom1(Int) ⊆ Dom1(int) ⊆ User = Dom1(use). The relation Use is defined as follows: DEFINITION 6 (Using types): u Use e iff ∃e ∈ e ∃o ∈ Ont (use(u, e) = o). According to this definition, the user u Uses the wfe-type e iff the user u uses a wfe-token of the type e to refer to some referent. The definition of relation Int is dual to the definition of the relation Use, DEFINITION 6i (Interpreting types): u Int e iff ∃e ∈ e ∃o ∈ Ont (int(u, e) = o). and it says that the user u Interprets the wfe-type e iff the user u interprets a wfe-token of the type e as some interpretandum. Because int ⊆ use, i.e. the relation int of interpreting tokens is included in the relation use of using tokens, the relation Int of interpreting types is included in the relation Use of using types (see Corollary 5a); however, from the Axiom given above for the relation Int for types, it follows that the user who Uses a type does not need to be the one who Interprets it. 262 A Logical Conceptualization of Knowledge on the Notion of Language... Because communication C(e) by means of the type e is a nonempty set, the above definitions lead to Corollary 5b) and the comment found at the top of this subsection is justified: COROLLARY: 5 a) Int ⊆ Use. b) int = use ⇒ Int = Use. c) 〈s, e, r〉 ∈ C(e) ⇒ s Use e ∧ r Int e. d) Use 6= ∅, Int 6= ∅, Point d) of the above corollary immediately follows from point c). 4.2.3. Problem of Adequacy of Language Communication Adequate, effective, successful communication in a community of Users by means of the expression-type e is based on the agreed meaning μ(e) of the expression-type e used by users who are senders of tokens of e in acts of communication, and based on the correlation μ(e) with the interpretation ι(e) of the expression-type e interpreted by users who are recipients of these tokens in the acts (cf. Wybraniec-Skardowska 2015). Compatibility of the meaning and the interpretation of the expression-type e leads to understanding between senders and recipients (see Diagram 7). Diagram 7 263 Urszula Wybraniec-Skardowska A disagreement between the meaning and the interpretation of the expression-type leads to misunderstanding, while ignorance of the interpretation of the expression-type leads to non-understanding. 4.2.4. Notions Relating to Language Communication It is obvious that the conceptual apparatus of the theory TLC has to be enriched by notions concerning meaning and interpretation of language expression-types. As we said before, these notions will be characterized in relation to the understanding of meaning as a manner of Using (usage) expressiontypes and interpretation as a manner of Interpreting (Int) these expressions; these manners are in a way built into this meaning and this interpretation, respectively. Interpretation indicates the meaning or meanings of a given expressiontype and cannot be identified with its meaning. Let us also note that the notion of interpretation does not need to be connected with sign-based systems of communication only; in semantics, it plays a special, central role. The notion of meaning is defined by means of the relation ∼= of having the same manner of Using wfe-types and the notion of interpretation – by means of the relation ∼=i. of having the same manner of Interpreting (understanding) wfe-types (see Wybraniec-Skardowska, 2007a, b; 2015, 2016). The definitions of these relations are as follows: DEFINITION 7 (having the same manner of Using types): e ∼= e′ iff ∀u ∈ User [(u Use e ⇔ u Use e′) ∧ ∧ ∀o ∈ Ont (∃e ∈ e (use(u, e) = o) ⇔ ∃e′ ∈ e′ (use(u, e′) = o))]. DEFINITION 7i (having the same manner of Interpreting types): e ∼=i e ′ iff ∀u ∈ User [u Int e ⇔ u Int e′ ∧ ∧ ∀o ∈ Ont (∃e ∈ e (int(u, e) = o) ⇔ ∃e′ ∈ e′ (int(u, e′) = o))]. Two wfe-types e and e′ have the same manner of Using (resp. of Interpreting) wfe-types if and only if every user of language L uses (resp. interprets) the other one every time he/she uses (resp. interprets) either of them, and every object is a referent (resp. an interpretant) of some token of the type e (used/interpreted by the user) iff it is a referent (resp. an interpretant) of some token of the other type e′ (used/interpreted by the user). The relation ∼=i having the same manner of Interpreting types is given if its arguments belong to Dom2(Int). So, we adopt the following axiom: 264 A Logical Conceptualization of Knowledge on the Notion of Language... AXIOM (domain of ∼=i): ∼=i ⊆ (Dom2(Int)×Dom2(Int)) ∩ ∼=. And, the relation ∼=i is a sub-relation of the relation ∼=, and it can easily be proved that it is a nonempty relation (from Corollary 5c: Int 6= ∅, and because it is a reflexive relation). THEOREM 1: The relations ∼= and ∼=i are equivalence relations in the set S∗. Definitions of meaning and interpretation of the wfe-type e are the following: DEFINITIONS 8 (meaning and interpretation): a) μ(e) = [e]∼= and b) ι(e) = [e]∼=i . The definition of interpretation ι(e) of the wfe-type e is dual to the definition of meaning μ(e) of the expression. According to these definitions: Meaning μ(e) and interpretation ι(e) of the wfe-type e is the equivalence class of all expressions possessing the same manner of Using or, respectively, Interpreting (understanding), as the expression e, and can be intuitively understood as a common property of all wfe-types having the same manner of Using or, respectively, Interpreting as the expression-type e. The property can be called the manner of using or, respectively, the manner of interpreting of the expression-type e. In this way, we are referring here to ideas originating from Ludwig Wittgenstein (1953) and Kazimierz Ajdukiewicz (1931, 1934), that is to understanding of the meaning as a manner of its Use/Interpreting. It is easy to see that we have: THEOREM 2 a) ι(e) ⊆ μ(e). b) int = use ⇒ ι(e) = μ(e). So, the notion of meaning is stronger than the notion of interpretation. 4.2.5. Dual Conceptual Counterparts It should be observed that the notions of the system: (*) use,Use,∼=, μ, have, within TLC, dual counterparts in the system: (**) int, Int,∼=i, ι. 265 Urszula Wybraniec-Skardowska All the notions of the system (marked with two asterisks) have dual definitions towards the corresponding definitions of the theory TLC concerning the notions of the first system (*). So, all theorems of the theory TLC formulated for the notions of (*) remain valid if we replace the notions of this system (*) with their dual counterparts of (**). The close relationships between the semantic-pragmatic notions of the systems (*) and (**) cause these notions to be often regarded as identical. However, each relation or function of the system (**) is only a sub-relation of its counterpart in the system (*) and not of all theorems of TLC concerning the notions of this system which have their dual counterparts. The meaning μ(e) of a wfe-type e and the interpretation ι(e) of the type e may differ. If that is the case, the communication C(e) by means of the wfe-type e does not have to be adequate. Using the notions ofmeaning and interpretation we can define the notion of adequacy of language communication. 4.2.6. Adequacy of Language Communication As it has already been mentioned, in language communication, interpretation indicates the meaning or meanings of the expression-type which intermediates in this communication. An expression-type may have more than one meaning. If it has more meanings, they are determined by subtypes of the expression, as for example, for the terms: 'key' or 'bank'. We will adopt the following definition of adequacy of communication: DEFINITION 9 (adequacy of language communication): If e has n (n ≥ 1) meanings determined by its subtypes e 1 , e 2 , . . . , en then C(e) is an adequate communication iff ∀k = 1, . . . , n (ek has determined interpretation and ι(ek) = μ(ek)). From the definition of adequacy of communication by means of wfe-type we obtain some conditions of adequacy of language communication: COROLLARY 6: a) If e has n (n ≥ 1) meanings determined by its subtypes e1, e2, . . . , en, then C(e) is not an adequate language communication iff ∃k = 1, . . . , n (ek does not have a determined interpretation or ι(ek) 6= μ(ek)). b) If e has an established meaning and e has a determined interpretation then C(e) is an adequate language communication iff ι(e) = μ(e). 266 A Logical Conceptualization of Knowledge on the Notion of Language... c) If e has an established meaning and e does not have a determined interpretation, then C(e) is not an adequate language communication. d) If e has an established meaning, e has a determined interpretation and ι(e) 6= μ(e)), then C(e) is not an adequate language communication. We see that the accord of meaning and interpretation is a necessary condition of adequate language communication by means of expression-type of L. The two next theorems provide us with some sufficient conditions for adequacy of communication by means of types. THEOREM 3: If int = use and e has an established meaning and a determined interpretation then C(e) is an adequate language communication. The above theorem follows from Theorem 2b and Corollary 6b. 5. Summary The main objective of the work presented was to provide a conceptual apparatus of a general logical theory of language communication. The outlined axiomatic 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 they require 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. However, so far it has not been possible to theoretically capture the intuitive relationships between the adequacy of language communication and the correctness of its communication acts. The paper is only an attempt at providing a conceptual apparatus for the theory. One cannot expect it to offer strong theorems as yet, although it seems that the theorems concerning the relationships between adequacy of language communication and adequacy of its communication acts should function well enough. 267 Urszula Wybraniec-Skardowska All the general conditions of adequacy of language communication discussed in the presented paper were shown as if they were 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. Finally, it can be noted that the conceptual apparatus of the theory can be enriched through the introduction of notions concerning some specific forms of communication, such as discourse and dialog. 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