On Universal Grammar and its Formalization Urszula WybraniecSkardowska Please note: Due to the complexity of the formatting of this paper, several dozen images and additional fonts were employed. If your browser does not support images or the additional fonts (e.g., Symbol), this paper will not display properly. In such cases, you may wish to download the Microsoft Word version of this paper.

1. General characteristics of the categorial approach to syntax At the beginning of this paper, to avoid misunderstanding, we explain that considerations concerning universal grammar should be understood as a theoretical and very general, formallogical approach to the logic of language. The paper treats logical syntax and logical semantics, conceptualized as a theory providing general principles for generating languages from the socalled classical categorial grammar. In this section we outline some universal ideas of such a categorial approach. 1.1. Some ideas concerning logical syntax of language The first ideas pertaining to the formalization of language syntax appear only in the Twentieth Century and have been provided by Rudolf Carnap (1934). The term 'logical syntax' introduced by Carnap is understood here in the narrower sense, namely as a field of the logic of language to which belong issues connected with the classification of expressions and their syntactic structure. We are mainly interested in the formalization of the categorial approach to logical syntax, which can be regarded as a formal theory of logical syntax elaborated in the spirit of the Polish tradition, which is here associated with: (1) LeśniewskiAjdukiewicz's theory of semantic categories, known today as the theory of syntactic categories, and (2) Tarski's axioms for metascience. As for theory (1), it was built by Leśniewski, not without the influence of Husserl's idea of pure grammar (19001901), for the languages of his protothetics (1929) and ontology (1930) systems. Ajdukiewicz (Die syntaktische Konnexität, 1935) significantly improved it with the help of indexassignation. LeśniewskiAjdukiewicz's theory can be regarded as a theory for the classification of linguistic expressions and takes into consideration some factors according to which compound expressions should: a) have unique categorization, like simple words; b) have functorargument structure; c) be reducible to atomic categories; d) satisfy the principle of syntactic connection; and (e) satisfy the rule of substitutability. LeśniewskiAjdukiewicz's theory is not, however, a formal theory, while Tarski's axiomatic approach to metascience (2) — which can be found in his famous paper on the concept of truth (1933) — provides the first formal foundations of metascience and thus also metalanguage in accordance with Leśniewski and Ajdukiewicz, Łukasiewicz and Post's ideas. This approach lay the groundwork for the first axiomatic, deductive base for language syntax and recursive grammar. In order to present such a grammar, Tarski gives the first deductive theory of strings in which the relation of concatenation of strings is a primitive concept characterized by axioms and serves to generate concatenations of strings from the vocabulary of a given language. This theory points a way for a formalization of the general theory of classical categorial grammar used by the first author. The theory of WybraniecSkardowska (1985, 1991, 1998) not only explicates certain ideas of the numerous researchers of the theory of syntactic categories including Bocheński (1947), Hiż (1960, 1967, 1968), Kubiński (1960), Geach (1971), Cresswell (1973, 1977), Marciszewski (1977, 1988), van Benthem (1984, 1988), Buszkowski (1989, 1997) but as an axiomatic theory it contains a comprehensive formal formulation which adheres to the original assumptions of approach (1). They are slightly different than in BarHillel (1950) and Lambek (1958, 1988) or in the combinatory logic of Curry (1961, 1963). Let us observe that although BarHillel coined the term 'categorial grammar' neither he nor his coworkers took into account the mentioned aspects (a)  (e) of the theory of syntactic categories because it has been adopted for computer language orientations; the syntactic description of language expressions has been replaced by the mechanical procedure of determining syntactic structures. WybraniecSkardowska's theory can be regarded as a formal theory in the certain sense of a universal grammar referring to the categorial approach. It will be denoted by TCG and formalize very generally both approaches (1) and (2) to categorial syntax and gives new proposals to categorial semantics. The syntactic researches initiated by the first author were inspired by her teacher, Jerzy Słupecki, a representative, like the famous Alfred Tarski, of the Warsaw Logical School; and the semantic ones were inspired by Suszko (1958, 1960, 1964), Lewis (1970), Montague (1970, Stanosz and Nowaczyk (1976), van Benthem (1984,1986, 1988), Buszkowski (1989) and Andrzej K. Rogalski. Let us however notice that any way to a formalization of semantic problems was opened by Tarski's famous paper (1933) on the notion of truth. 1.2. Certain conventions pertaining to language Prior to presenting general ideas of the theory of categorial grammar it seems indispensable to define the following conventions:
Such characterization of language is essential because:
1.3. Intuitive foundations of TCG Let us consider the intuitive background of the theory TCG as a theory of categorial syntax, and the same theory of categorial grammar. A strict and axiomatic presentation of the theory in the space of this paper is impossible. Thus we refer the reader to the book (1991) and paper (1989) of the first author (cf. also1998). The simplest syntactic characterization of any language L gives the following ordered system: (L) < U_{L}, ~, V_{1}, c, W_{1}; S > consisting of: U_{L}  the universe class of all linguistic objects (broadly understood inscriptions); ~_{ }  the indiscernibility (equiformity) relation in U_{L} given on the tokenlevel; V_{1}  the vocabulary of all simple words as a subset of U_{L}; c  the ternary relation of the concatenation on U_{L} to generate from V_{1}: W_{1}  the set of all words of which a subset is the set: S  the set of all wellformed expressions (for short wfes). As we mentioned, all the notions of the discussed theory TCG are understood very generally. Regardless of whether the tokenlevel or the typelevel is considered as the primitive level of formalization of language, we assume that in TCG the universe U_{L} and the vocabulary V_{1} are primitive notions. The same pertains to the relations of indiscernibility ~ and concatenation c. We assume axiomatically that V_{1} is a subset of U_{L}, and V_{1} and U_{L} are nonempty sets. The vocabulary V_{1} can be exactly established and closed as e.g. in formalized languages, or open as e.g. in natural languages. The relations of indiscernibility ~ and concatenation c for tokens are understood very broadly too. If we apply them to types, then indiscernibility is a simple identification relation and concatenation c is a settheoretical function of juxtaposition of two types in a new one. On the tokenlevel, the relation of indiscernability is characterized axiomatically as an equivalence relation. Intuitively it can, but does not have to be, the relation of empirical similarity if the pragmatic aim is, for instance, to compose an advertisement or a special program/invitation, as we can see in Example 1 regarding the different token words "Boston". The relation of concatenation c considered on the tokenlevel may, but does not have to be, understood as a sequencing of two similar tokens, if the pragmatic aim is the same (see Example 1). Intuitively, concatenation of two written tokens A and B, for example an English one (resp. Semitic one), is written token D* that is made up by adding to A*, indiscernible from A, on the right side (resp. on the left side), the written token B*, indiscernible from B. The concatenation can (see Example 1) also be a relation of the nonlinear juxtaposition of two tokens, which thus form a single token, e.g. in the hieroglyphic script and mathematical formulas. We assume two axioms stating, respectively, that: concatenations of two pairs of tokens with the first and second elements pairwise indiscernible yield two indiscernible tokens; and that a token which is indiscernible from the concatenation of two tokens is also their concatenation. In this way the relation of concatenation defined on tokens is not a settheoretical function. Example 1. This example shows the functioning of the relation of indiscernibility and the relation concatenation in concrete cases. We see that the following quadruples of tokens: A, A', A'', A'' ; B, B', B'', B'' and C, C', C'', C'' are indiscernible, and concatenations of tokens A and B and C are different but indiscernible from the tokens words "Boston Symphony Orchestra". As for the set W_{1} of all words, it is defined as the smallest set of objects of U_{L} containing the vocabulary V_{1} and closed under concatenation c. The abovegiven syntactic characterization of any language L is typical for any formal grammar, but if we want to describe any language in formal theory we must define the notion of the set S of all wellformed expressions. This is the basic task of the formal theory of categorial syntax axiomatically characterized from a universal point of view. In order to define this notion on the ground of the theory TCG, the formal characteristics requires the consideration of an ordered system more complex than (L), denoted here by (L^{k}) (k = 1, 2): where k = 1, when language is characterized on the tokenlevel as the language of token objects, and k = 2 for language characterized on the typelevel as language of type objects. The sign "=^{1}" denotes here the relation ~ of indiscsernibility among tokens while the symbol "=^{2}" — the ordinary relation = of identification of types. In (L^{k}) beside notions which correspond to the notions of the system (L) we distinguish the following primitive or defined notions of TCG: If we want to build the theory TCG as the theory of a description of all notions of the system (L^{k}) (k = 1, 2), on two levels, we have to establish what is the first level of formalization of language L. If we begin with the formalization on the tokenlevel then the terms listed in the system (L) and taken with the superscript 1 have already been characterized. The remaining terms in (L^{1}) are new: primitive or defined ones of the theory TCG, and all dual notions of the system (L^{2}) pertaining to types are defined on the typelevel. Conversely, if we begin with the formalization of the language of types, then we first characterize, in a similar way as before, the notions of system (L^{2}) on the typelevel and then the dual notions of system (L^{1}) on the tokenlevel. We will only outline an intuitive understanding and characterization of the terms occurring in (L^{k}) which do not belong to (L) without taking into consideration, seperately, the tokentype distinction. We can find a formal presentation of the theory TCG in WybraniecSkardowska (1989, 1991). The set S of wfes is defined by means of two sets: (of all complex expressions) and (of all simple expressions). It is important that in distinguishing these sets from W_{1} we use categorial indices, which play a principal role in approach (1). Categorial indices are some objects of the universe U_{L} (on the tokenlevel they are tokens and on the typelevel they are types) but they do not belong to the set W_{1} of words of a given language L, but are rather words of the metalanguage of that language. They are socalled auxiliary words of L belonging to the set W_{2} of all auxiliary words of L. The set W_{2} is generated from the nonempty, auxiliary vocabulary V_{2} of L (a further primitive notion of TCG) by means of concatenation relation c, similarly to the set W_{1} of words from the vocabulary V_{1}. The vocabularies V_{1} and V_{2} are, of course, disjoint sets. The auxiliary vocabulary V_{2} includes basic indices and auxiliary symbols, e.g. brackets, commas, fraction lines, etc. The set of all auxiliary words W_{2} includes two nonemty and disjoint sets: the set of all basic indices from V_{2} and the set of all functoral ones. The sum and gives the set E_{2} of all categorial indices. Functoral indices are formed from basic ones by the relation r_{2} of the formation of functoral indices (another primitive notion of TCG), and the set is defined as its counterdomain D_{2}(r_{2}). In the theory TCG the relation r_{2} of the formation of functoral indices is characterized by axioms; formally it is a binary relation but it can be regarded as any finite, at least ternary, relation defined on subsets of W_{2} which are sets of indices of words of L. It can be treated as a substitute for any rule for the formation of functoral indices of independently of their specific notation, in particular: fractional, quasifractional, parenthesis (see Example 2). Example 2. Let us assume that for a given language L^{1} its auxiliary vocabulary is equal to 1) { s, n, /, ' } or to 2) {s, n, /, \, (,)} or to 3) {s, n, ' , (,)}and the set of all basic indices of L^{1} equals {s, n}, where s is the index of sentences and n is the index of names of L^{1} and concatenation consists in rightsided linear juxtaposition. The relation r_{2} replaces any rules which allow us to form from two indices a new, functoral index as well as any rules which make it possible to create a new functor from three or more indices. Certain of them can serve to form indices 1) by means of the auxiliary sign "/" — slash (see Ajdukiewicz 1935) and others 2) with both "/" and "\" — backslash (see Lambek, 1958) or in the case 3) by means of coma "," and parentheses "(" and ")" (see van Benthem, 1986). For instance, the following different indices of a quasifractional or parenthesis form assigned to the same kind of functors can be created by means of suitable rules corresponding to the relation r_{2}:  indices of a sentenceforming functor of one name argument 1) s/n, 2) s/n or n\s , 3) (s, n);  indices of a sentenceforming functor of one argument which is a sentenceforming functor of one name argument 1) s//s/n , 2) s/(n\s) or (s/n)\s , 3) (s, (s, n));  indices of a sentenceforming functor of two name arguments 1) s/nn , 2) n\s/n , 3) (s, nn);  indices of a functor forming a sentenceforming functor of one name argument and whose arguments are two functors of the same category 1) s/n//s/n, s/n , 2) s/n\\s/n//s/n , 3) ((s, n), (s, n)(s, n)). For a fixed language L a concrete application of the rule r_{2} is its use as a quasifractional notation: a / b_{1} b_{2} ... b_{n} for the functoral index of any
functor forming any expression with the categorial index a and of n arguments
(n 1) which are, in turn,
expressions with indices: r_{2}(a, b_{1},_{ }b_{2}, ... b_{n}; b). The unique categorization (see a)) of defined words of L is obtained by the relation i of the indication of indices to words, which is a new primitive notion of TCG. The relation i assigns to every word of a subset of W_{1} one (with exactitude to indiscernibility) categorial index from E_{2}. It is only from the set of words possessing categorial indices that we separate the set E_{1} of all expressions of categorial language L as the sum of two disjoint sets of expressions: the set of words of the vocabulary V_{1} possessing indices ( is the intersection of V_{1} and the domain D_{1} (i ) of the relation i , similar as the set is the intersection of V_{2 }and the domain D_{2} (i ) of i ), called the proper vocabulary of language L or the set of all simple expressions of L, and the set of all its complex, functorargument expressions which is the counterdomain D_{2}(r_{1}) of the relation r_{1} of the formation of complex, functorargument expressions. While for any language L these complex expressions are determined by the specific syntactic rules of L, in theoretical considerations these rules are replaced by the relation r_{1}. The relation r_{1} is a primitive notion of the theory TCG. It is a binary relation but it can be regarded as any finite, at least ternary relation defined on sets of words possessing indices, i.e. defined on any finite number > 2 of domains D_{1}(i). And what is very important is that relation r_{1} can be treated as a substitute of any rule of forming functorargument expressions of L. The relation r_{1} refers to syntactic rules of any categorial language L without regard for the notations, symbolism, or calligraphic system used to form their complex expressions (see Example 3). Example 3. Let us assume that we formalized the classical logic in such different ways that functors of implication and conjunction, and quantifiers have, respectively, the symbols from the proper vocabulary : and concatenation consists in rightsided linear or nonlinear juxtaposition. The relation r_{1} replaces various rules allowing us to form from at least two expressions a new one. These rules can be rules of the formation of terms or sentential expressions of classical logic. Certain of them can serve to form sentential functorargument expressions, which are different but synonymous expressions in various formalized languages of the classical logic. For instance, we can form the following three synonymous implication expressions with three kinds of connectives and quantifier symbols 1) or 2) or 3): We see that the above implications recorded by means of different symbols have the same meaning and can be created by means of different suitable syntactic rules corresponding to the relation r_{2} ( the third of them is written down in Łukasiewicz's parenthesisfree notation). If the relation r_{1} holds among the system of expressions (f, e_{1}, e_{2}, ..., e_{n}) and the expression e, where f is the main functor of e and e_{1}, e_{2}, ... , e_{n} are its n (n >0) arguments, and the expression e is formed by means of this functor and its arguments, then on the ground of theory TCG we record this fact as follows: r_{1}( f, e_{1}, e_{2}, ..., e_{n}; e ). For a fixed language L instead of this formula we can use the following nonformal notation (on the typelevel): and if e is a wfe and has the index a, then the index b of its main functor f is formed from the index a and indices b_{1}, b_{2}, ..., b_{n} of arguments of this functor, respectively, and has to satisfy the principle (r) given below, and so the following formula of the theory TCG: r_{2} (a, b_{1}, b_{2}, ... , b_{n} ; b). We can use for it (on the typelevel) the mentioned nonformal notation: The set S of all wfes of L can be generated by the ordered system in which (r) is a rule establishing relationships between the index of any functorargument expression and the index of its main functor and indices of its arguments. We call it the principle of the syntactic connection of compound expressions. For any expression e in the form (e) it has the following verbal formulation:
The system (CG_{L}) may be regarded in TCG as a general reconstruction of any classical categorial grammar whose idea goes back to Ajdukiewicz (1935), and in the formallogical characteristics of language one can also find some ideas of other formal grammars, e.g. generative grammars (Chomsky 1957). The system (CG_{L}) generates any language L described by the theory TCG. The most important notion of the theory TCG is the set S of all wfes of L. It is defined as the smallest set containing all simple expressions from and every such functorargument expression of which has the property that both it and any compound expression being its constituent satisfy the rule (r). This definition allows us (see d) and c)) to describe an algorithm for testing the syntactic connection of expressions of L (if L is a categorial language without variables and operators that bind them). Let us know that the notion of the set S of all wfes of L could be also formally introduced in a little modified way if L is any categorial language that include operators and variables bound by them. A narrow space of this paper does not allow to concentrate on this issue; thus we refer the reader to the books of the first author (1985, 1991). The categorial approach to language L is connected with the unique categorization (see a)) of each of its expressions, i.e. with assigning it to a defined syntactic category which in TCG is defined as a class of expressions possessing indiscernible indices. The syntactic category of expressions possessing the index x Î E_{2} is denoted by_{ }CATx and Ct( S) is the family of all syntactic categories of expressions of S. It simultaneously is a logical partition of the set S. The family Ct( S ) is divided into two families: the family of all basic categories, i.e. categories with basic indices from , and the family of all functoral categories, i.e. categories with functoral indices from . The sum of all basic categories of the first family gives the set B of all basic wfes of L and the sum of all functoral categories — the set F of all functors of L. The sum of sets B and F is equal to the set S, and B and F are disjoint sets. So, symbolically: We mention in this place that the traditional definition of syntactic categories drawing upon the notions of replaceability (see e)) and a wellformed expression, in particular a sentence, requires a prior definition of these notions in order to avoid the risk of a vicious circle, so the notion of a relation of replaceability should be first defined. It is possible on the base of the theory TCG and leads to theorems which are important for the theory of syntactic categories (see WybraniecSkardowska 1989, 1991). The scope and nonformal character of this paper justify omitting these issues. Let us also observe that the unique categorization and unique functorargument structure of linguistic expressions is not idealization if we treat them functionally and use them in contexts in accordance with pragmatic aims. Syntactically or semantically ambiguous expressions can, in particular, be understood as schemas of wfes with one functorargument structure and with one (with exactitude to indescernibility) categorial index. 2. Two opposite ontological approaches to logical syntax As we know, language generated by the system (L^{k}) (k = 1, 2) should be characterized on two levels which refer to the distinction tokentype made by Peirce (19311935) and used by Carnap as signevent and signdesign (1942). The twofold ontological character of linguistic objects understood as tokens (material objects) or types (abstract objects) should be emphasized in the formalization of theory TCG. The choice as the first level of formalization of either tokens or types is in relation with two fundamental strains in philosophy: nominalistic (materialism) and Platonistic (idealism), which formed in the controversy over universals. If we begin our formalization of the theory TCG (as a theory T1) from the tokenlevel and we first characterize language L^{1} and the notions of the system (L^{1}), i.e. sets of tokens and relations defined on such sets, and then we formally describe language L^{2} expanding the theory T1 by introducing the concepts of the system (L^{2}), i.e. the sets of types and relations defined on such sets, then we present a nominalistic approach to the categorial syntax (see WybraniecSkardowska 1985, 1991). At the typelevel, all notions of system (L^{2}) in such extended theory are derived constructs defined by means of the dual notions at the tokenlevel. Types are obviously defined as equivalence classes of tokens by means of the relation =^{1} of indiscernibility (the relation ~). Every set of types, which is a dual counterpart of a set of system (L^{1}), is defined by means of the dual set of tokens and the relation =^{1} of indiscernibility of the system (L^{1}). Similarly, every relation between types characterizing language L^{2}, on the typelevel, is defined by means of the dual relation between tokens and the relation =^{1}. It is possible to present another biaspectual formalization of the theory TCG (as a theory T2) that embraces a different, opposite and categorial approach to language syntax — the Platonistic approach. It assumes that the first level of such formalization is the typelevel and the second one is the tokenlevel. First, on the typelevel, the theory T2 is constructed as a theory characterizing language L^{2} and the notions of the system (L^{2}), i.e. types, appropriate sets of types and relations between them. The second level of formalization of T2 concerns the tokenlevel and is considered in the dual theory as the theory describing the notions of the system (L^{1}) and, in this way, language L^{1}. The axioms and definitions of T2 at the typelevel are either dual analogous counterparts of the axioms and definitions of T1 or expressions equivalent to the latter. In T2 we assume axiomatically that types are nonempty sets and two types are equal (are indiscernible) if some element belongs to both of them. Two of the basic notions at the tokenlevel, i.e. a token and the relation of indiscernibility, are characterized as follows: a token is an element of a type, and two tokens are indiscernible if and only if they both are the elements of a certain type. All remaining notions of the system (L^{1}) are defined by appropriate dual concepts from the typelevel. It has been proved that both dual approaches to language syntax, nominalistic and Platonistic, are theoretically equivalent because both theories T1 and T2, formalized on the two different abovementioned levels and on the ontologically opposite base of these levels, are inferentially equivalent (see WybraniecSkardowska 1988, 1989). In the next section we will introduce some basic semantic notions without regard for the two abovementioned manners of formalization of the theory TCG as the theory T1 or as the theory T2. 3. Categorial semantics The biaspectual formalization of language generated by categorial grammar and thus the formalization of theory TCG on the two levels is important but not sufficient. This is because the ability to use language requires the same knowledge of the users of language about interpretation of its wellformed expressions, in particular sentences, and when these sentences are true. It is not a trivial task to provide a general definition of the truthfulness of any sentence of any language generated by the grammar (CG_{L}). In this section we try to outline a certain basis for accomplishing this task and to introduce some theoretical, formallogical background for categorial semantics. For this aim we have to expand the theory TCG (cf. WybraniecSkardowska, 1998). It should be developed on the typelevel for language L^{2} characterizing an extended system (L^{2}) regardless of its formalization as T1 or T2. Speaking about semantic questions of language L^{2} we will omit all superscripts 2 relevant to it. 3.1. On the adequacy of syntax with respect to semantics In what way do we want to understand interpretation of wellformed expressions of any categorial language L? Let us proceed to try to outline some base for categorial semantics by describing an intuitive side of theoretical consideration. We know what an interpretation of a wfe, in particular a sentence, is if we can assign to it — and earlier, where present, to each of its well formed constituents — an object to which the expression refers or which this sentence represents. This object belongs to an ontological category and is determined by a settheoretical function of denotation which assigns an object of reality described by L to every its wfe, in particular to each sentence of L. This object is called the reference or denotation of this expression. So, semantic interpretation consists in defining for wfes of L a denotation function, which can also be called a reference function or an extension function. The choice of a given denotation function for wfes of L is limited to the ontology for L. Only when we know what kind of ontology we are dealing with can we assign ontological categories to objects, which are described by L and are the values of the denotation function. In our categorial approach to semantics we take into consideration the referential aspects. So, the theory TCG needs to be enriched by semantic and ontological notions. It is connected with a certain expansion of the system (CG_{L}) characterizing in a general way any categorial grammar generating any language L. This expansion consists in adding to (CG_{L}) the notion of the denotation operation, which will replace any denotation function defined for any language L. We develop TCG according to some innovative ideas of Frege, which are visible in the syntactic and semantic categorial agreement of language expressions, i.e. in the principle (CA) of categorial agreement (see Suszko, 1958, 1960; Stanosz and Nowaczyk, 1976, Buszkowski 1989) based on (CA) the agreement of the syntactic category of each language expression with the ontological category assigned to the reference of this expression. This principle is a general rule of interpretation in the ontology of any categorial language generated by the grammar (CG_{L}). Keeping this principle we keep a correspondence between categorial syntax and referential, categorial semantics: every two expressions belonging to the same syntactic category have references (denotations) belonging to the same ontological category, and conversely. This correspondence can be called the adequacy of syntax with respect to semantics. It allows us to identify syntactic categories with semantic categories, where these latter are understood as sets of expressions whose denotations belong to suitable ontological categories. Let us note that categorial indices, used first by Ajdukiewicz (1935) to characterize the syntactic role of expressions in sentences, were used first by Suszko (1958, 1960) as a tool for the coordination of expressions and extralinguistic objects. The authors will use them in the same way in the next subsection, which will be presented more formally than the preceeding. 3.2. On typical ontology and denotation operation The semantic description of any categorial language L requires defining an appropriate ontology. Speaking about an ontology we will take into account the family of sets determined by categorial indices of E_{2} and called ontological categories. The latter correspond to syntactic categories with the same indices, which serve as a tool for the coordination of expressions of language L and their references, which belong to the reality described by language L. The ontological categories create a branched hierarchy determined by indices from E_{2}, like the syntactic categories. The ontological category with the index x will be denoted by ONTx. It is a set (not necessarily a nonempty set) consisting of settheoretical objects constructed by means of universes of the reality described by L and determined by a basic index b (b Î ). Universes are primitive notions of the extended theory TCG. We assume axiomatically that for each b Î the universe is a nonempty set, i.e. i.e. the ontological category with (see notation (fi)) the functoral index a / b_{1} b_{2} ... b_{n} is the set of all settheoretical functions from the Cartesian product ONTb_{1} ´ ONTb_{2} ´ ONTb_{n} of ontological categories with indices, in turn, b_{1}, b_{2}, ... , b_{n} , into the ontological category ONTa. D2. By reality corresponding to L we will understand the set ONT_{L} defined as the sum of all ontological categories from typical ontology which have indices from E_{2} and, at the same time, are assigned to the wfes of S. So, symbolically (cf. (Sî þ) and (SÈ )): i.e. the reality ONT_{L} is the sum of two disjoint sets: the set ONT_{B} of all ontological categories with basic indices corresponding to basic wfes of the set B and the set ONT_{F} of all ontological categories with functoral indices assigned to functors of the set F. As we mentioned, in theoretical considerations the counterpart of every denotation function is the denotation operation d. It is understood as a substitute for any concrete denotation function for fixed language L. It is a new component for the semantical formal characteristics of categorial grammar as the following system: In this way the set S of all wfes of any interpreted language L will be generated by the categorial grammar <(CG_{L}); d >. According to the ideas of Frege's semantics, the mutual dependence of syntactic and semantic characteristics of L should be considered by keeping the principle (CA) of categorial agreement. Denotation operation d is a new primitive notion of the extended theory TCG. It is characterized by the following axioms: (d maps the set S of all wellformed formulas into the reality ONT_{L} corresponding to L ), (the axiom A2 corresponds to the principle (CA) of categorial agreement: any wellformed expression belongs to the category with the index x if and only if the reference d(e) of this expression belongs to the ontological category with the same index x ). where f Î F is the main functor of e and e_{1}, e_{2}, ... , e_{n }are its arguments, the following condition of a homomorphism holds: (the reference of the expression e is the value of denotation of its main functor defined on references of successive arguments of this functor). So, on the basis of TCG from the condition 2^{0} of the definition of the ontological category with the functoral index (see D1, 2^{0}) and the above axiom A2 we have, according to the Frege's idea, that the reference of the main functor f Î F of the compound expression e of the set S is a function belonging to the set ONT_{F }and it is defined on the denotations (references) of successive arguments of this functor. Thus the condition (h) is a correct formulation of the condition of homomorphism because we treat functors as partial settheoretical functions defined on expressions of L, values of which are expressions of L, too. So, the denotation operation d should be understood as a mapping that is a homomorphism of a specific algebraic language structure into an algebraic ontological structure. Then we can formally define the concept of a model of categorial language L and the notion of the truthfulness of its sentences. We will do this in the next subsection. 3.3. Models of categorial language Let us consider the subset F^{s} of the set S of all wfes of categorial language L such that where the proper vocabulary is the set of generators of the partial algebra L, and D5. The algebraic ontological structure of reality is understood as the following partial algebra: where the set ONTF^{s} is composed of all functions of ontological categories determined by indices of simple functors of F^{s} and the set is the set of generators of the partial algebra R_{L} and consisting of all ontological categories determined by indices assigned to all the simple expressions of the vocabulary . So, really, the denotation operation d as satisfying the condition (h) of the axiom A3 is a homomorphism from the algebra L into the algebra R_{L}. D6. A model of categorial language L we call the substructure consisting of the homomorphic images of the sets of the algebra L with respect to the denotation operation d. The operation d is an epimorphism from L onto M_{L} . D7. The model M_{L} is called a proper model of language L if the denotation operation d is a onetoone operation. Then it is an isomorphism from L onto M_{L}. Let us observe that the notion used here of a model of language is a referential model and differs from the standard notion (cf. Hodges, 1993). 3.4. On the notion of truth Introducing the notion of true sentences of language L requires us to mark out the category of sentences. For this reason in the set of basic indices we have to distinguish a sentence index sen, assuming that Then accepting the convention that D8. CAT_{sen} is the syntactic category of sentences of L we can axiomatically assume that i.e. that there exist basic wfes of L which are its sentences. Let The set Sen we call the set of all sentences of language L. By means of the denotation operation d we can define the notion of a true sentence of language L. For this aim we distinguish as the chief ontological category ONT_{sen} = U (see D1, 1^{0}, A4) as a universe which satisfies the following axioms: where T and F are new primitive notions of the theory TCG which we can intuitively understand as truth and falsity, or the set of all states of things that are and the set of all state of things that are not, respectively. Because B Í S ( see (SÈ )), from D9, A6 and A2, (CA) immediately follows that So, we see that by means of the denotation operation d we can introduce the definition of the notion of a true sentence in the following way: D10. If e Î Sen and M_{L} is a model of L then e is true in M_{L} iff d (e) Î T in M_{L}. (L = L^{2 }). The definition D10 says that e is a true sentence in a model M_{L} iff its reference in this model corresponds to truth. As well as saying that e is true in M_{L} we can also say that M_{L} is a model of the sentence e. For formalized languages of the zero and first order and for a denotation function satisfying the axioms of the theory TCG (see WybraniecSkardowska, 1998), the question of whether the notion of satisfaction in Tarski's sense can be replaced by the notion of denotation appears. This problem is being solved by the second author of this paper. The reader might have a question whether the biaspectual formalization of the theory TCG described in Section 1 would also be useful to semantic problems. The answer must be positive because, apart of the semantic definition D10 of the notion of truth, we can also introduce a syntactic definition of this notion. Such a definition can be given as the following counterpart of the classical definition of truth: D11. If e is a typesentence of L^{2} (e Î Sen) and e Î e (e is a tokensentence of L^{1} ) then It states that: if e is a typesentence with the representative e , which is a tokensentence, then e is a true sentence if and only if e ; the definition D11 can be interpreted in the classical way as follows: e is a true sentence if and only if the state of things is as the representative e of the sentence e says that is. Let us observe that the typesentence e in D11 can be understood as the equivalence class [e ]~ and simultaneously as the quotation name 'e ', and in this way, this definition corresponds to Tarski's famous convention (T) (cf. Grzegorczyk, 1997). Let us note that if we introduce the abovegiven syntactic definition of a true sentence, then the denotation operation should satisfy the additional condition: D10^{1}. If e is a typesentence of L (e Î Sen, L = L^{2} ) then for any model M_{L} of L d(e) Î T in M_{L} iff e is a true sentence of L. In other words, on the base of D11 and keeping the conditions of D10^{1} the above equivalence can be replaced by the following: d(e) Î T in M_{L} iff e , i.e. the reference of the typesentence e corresponds to truth if and only if the state of things is as the representative e of sentence e says that is. On the base of the definition D10^{1} it can clearly be seen that both definitions of a true sentence, the semantic and the syntactic, are equivalent on the ground of the theory TCG, i.e. the following theorem holds: If eÎSen, then e is a true sentence of L iff e is true in any model M_{L} of L. The mutual relationships between the notions of truth, denotation and satisfaction for formalized languages of systems of knowledge are the subject matter of the authors' further researches. 4. Final remarks In accordance to the purposes of this paper, we have presented the main ideas connected with the formalization of classical categorial grammars and, thus, with the languages generated by such grammars. In the formal logical and categorial approach, we have taken into consideration both semantic and syntactic, as well as ontological aspects. The generality of the approaches given to such a formalization allows us to understand the discussed theory TCG as a theory of universal grammar. It seems to the authors that languages built according to other principles than those formulated in this paper, may be formal models to which the basic notions of this paper — considered in formal systems of categorial languages characterized both syntactically and semantically — may be applied. In the categorial approach to the syntax of language, the biaspectual character of language is important, and depends on the ontological status of its objects: firstly as language of token expressions, which are physical representations of types of expressions (at the tokenlevel); and secondly as language of typeexpressions (at the typelevel). The bilevel nature of language makes it possible to reconstruct a syntactic, classical definition of truth. Categorial semantics is referential, and departs from the classical semantic description orginated by Tarski (1933). Thus, here in this paper, the basic semantic notion is the notion of denotation, and not of satisfaction. By means of the notion of denotation we can give a semantical definition of true sentences. In this categorial approach, every linguistic expression has a reference. In particular, in formalized languages, names and individual variables should correspond to other references belonging to other ontological categories, and the same is also true in the case of sentences and sentential functions. Quantifiers can be treated as certain functions whose denotations are functions in the reality described by language. The assigning to the quantifiers and their denotations of suitable syntactic or, respectively, ontological categories depends on the number of variables, to which these quantifiers bind (see WybraniecSkardowska1998). However, the categorial approach proposed in the theory TCG omits the issue, which is frequent in practice, of the ambiguous assigning to linguistic expressions of constituents of reality corresponding to them; in the formal approach the authors do not consider the situational context. This approach also does not deal with the assigning of objects to tokenexpressions of language (at the tokenlevel). It is possible to develop this theory, taking into consideration these issues, as has been proposed by Dr. Edward Bryniarski, with whom the authors of this paper are conducting their further research. 
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