The first ideas pertaining to the formalization of language syntax appear only in the Twentieth Century and have been provided by Rudolf Carnap 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.
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.
They are slightly different than in Bar-Hillel and Lambek , or in the combinatory logic of Curry , Let us observe that although Bar-Hillel coined the term 'categorial grammar' neither he nor his co-workers 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because it has been adopted for computer language orientations; the syntactic description of language expressions has been replaced by the mechanical procedure of determining syntactic structures. Wybraniec-Skardowska'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.
Let us however notice that any way to a formalization of semantic problems was opened by Tarski's famous paper on the notion of truth. Prior to presenting general ideas of the theory of categorial grammar it seems indispensable to define the following conventions:. 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 and paper of the first author cf. The simplest syntactic characterization of any language L gives the following ordered system:. U L - the universe class of all linguistic objects broadly understood inscriptions ;.
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V 1 - the vocabulary of all simple words as a subset of U L ;. As we mentioned, all the notions of the discussed theory TCG are understood very generally. Regardless of whether the token -level or the type -level 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 vocabulary V 1 can be exactly established and closed as e. If we apply them to types, then indiscernibility is a simple identification relation and concatenation c is a set-theoretical function of juxtaposition of two types in a new one.
On the token -level, the relation of indiscernability is characterized axiomatically as an equivalence relation. The relation of concatenation c considered on the token -level 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. The concatenation can see Example 1 also be a relation of the non-linear juxtaposition of two tokens , which thus form a single token , e.
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 set-theoretical function. Example 1. This example shows the functioning of the relation of indiscernibility and the relation concatenation in concrete cases.
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 above-given 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 well-formed expressions.
This is the basic task of the formal theory of categorial syntax axiomatically characterized from a universal point of view. 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 begin with the formalization on the token -level 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 type -level. 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 type- level and then the dual notions of system L 1 on the token- level. 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 token-type distinction.
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 token -level they are tokens and on the type -level 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 so-called auxiliary words of L belonging to the set W 2 of all auxiliary words of L. The set W 2 is generated from the non-empty, 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. 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 counter-domain 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.
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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, quasi-fractional, parenthesis see Example 2. Example 2. 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.
For instance, the following different indices of a quasi-fractional or parenthesis form assigned to the same kind of functors can be created by means of suitable rules corresponding to the relation r 2 :. For a fixed language L a concrete application of the rule r 2 is its use as a quasi-fractional notation:. We will apply this non-formal notation later instead of the functoral index b, which satisfies the following formula of the theory TCG :. 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.
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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 , functor-argument expressions which is the counterdomain D 2 r 1 of the relation r 1 of the formation of complex , functor-argument 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.
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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. And what is very important is that relation r 1 can be treated as a substitute of any rule of forming functor-argument 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 :. 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.
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Certain of them can serve to form sentential functor-argument 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 :.
If the relation r 1 holds among the system of expressions f , e 1 , e 2 , For a fixed language L instead of this formula we can use the following non-formal notation on the type -level :. 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 , and in the formal-logical characteristics of language one can also find some ideas of other formal grammars, e. It is defined as the smallest set containing all simple expressions from and every such functor-argument 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 , The categorial approach to language L is connected with the unique categorization see a of each of its expressions, i. 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.
So, symbolically:. We mention in this place that the traditional definition of syntactic categories drawing upon the notions of replaceability see e and a well-formed 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 Wybraniec-Skardowska , The scope and non-formal character of this paper justify omitting these issues. Let us also observe that the unique categorization and unique functor-argument 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 functor-argument structure and with one with exactitude to indescernibility categorial index. 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.
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If we begin our formalization of the theory TCG as a theory T1 from the token -level and we first characterize language L 1 and the notions of the system L 1 , i. At the type -level, all notions of system L 2 in such extended theory are derived constructs defined by means of the dual notions at the token -level. It assumes that the first level of such formalization is the type -level and the second one is the token -level. First, on the type -level, the theory T2 is constructed as a theory characterizing language L 2 and the notions of the system L 2 , i.
The second level of formalization of T2 concerns the token -level 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 type -level 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 token -level, i. All remaining notions of the system L 1 are defined by appropriate dual concepts from the type -level. 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 above-mentioned levels and on the ontologically opposite base of these levels, are inferentially equivalent see Wybraniec-Skardowska , In the next section we will introduce some basic semantic notions without regard for the two above-mentioned manners of formalization of the theory TCG as the theory T1 or as the theory T2.
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 well-formed 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, formal-logical background for categorial semantics.