2. The categorial framework and the well-formed formulae of the ternary description language

The first and the most important step in the construction of the formal apparatus, which fills the requirement listed above, is the choice of

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appropriate categorial framework. According to Stephan Korner, "To indicate a thinker's categorial framework is to make explicit: (i) his categorization of object, (ii) the constitutive and individuating principles associated with the maximal kinds of his categorizations, (iii) the logic underlying his thinking" (Korner, 1970).

Aristotelian logic was based on the categorial opposition: thing (subject)-property (predicate). The relational logic (De Morgan, Povarnin, Serruce) presupposed the oppositions: things-relation. Instead of the traditional form of judgment: "S est P" in that logic, the scheme aRb was accepted.

The classical predicate logic (Frege, Russell, Pierce) has separated the world into two categories: individuals and predicates. Here the distinction between properties and relations has been reduced to a pure quantitative one. A one-place predicate expresses a property, a two-place, a three-place, etc. predicate expresses a relation.

In our categorial framework there are three basic categories: Things, Properties, Relations. This accounts for the name of our formalism – the Ternary Description Language.

The constitutive and individuating principles associated with those categories are the following. Everything which can be named or described is a thing (= object = entity). For example – "love", "the struggle", "the identity". Everything which distinguishes one thing from another is a property, e.g., "red", "old". Everything which constructs one thing from other is a relation, e.g., "read", "marry." If "John reads a book," we have a pair which consists of John and a book. If "John gets married to Margaret," we have a new object – a family. Of course, old John is a new object in relation to John. But old John is John. Nevertheless the family is not John, as well as the family is not Margaret. That is the most essential difference between properties and relations. The numerical difference is not essential. John loves Margaret. And John loves himself. In both cases "loves" is a relation in spite of John who loves and John who is loved, is the same object.

Certainly, the definitions listed above are not rigorous. They presuppose the knowledge of such things as "named," "described," "distinguishes," "constructs." If we had defined them, we should use the terms "thing, property, relation". Therefore, we may be accused of the mistake called "circulus vitiosus". But that mistake is inevitable in the definitions of categories. For the reason given "Thing," "Property,"

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and "Relation" must be considered as primary, undefinable notions. Our "definitions" are, more specifically, elucidations which can be useful for understanding our conception of those categories.

Another essential feature of our conceptual framework is the con textual character of distinction between the categories. It means that a thing in one context can be a property or a relation in another context. For instance, in the sentence "Love is a good affection," the word "Love" expresses a thing (=an object, an entity). In the sentence, "That affection is love," "love" is a property. In the sentence, "John loves Margaret," the word "loves" denotes a relation.

Things, properties and relations can be, in their turn, definite, indefinite and arbitrary. With the help of these categories we reflect extra logical reality in an essentially different manner from that done by the categories of set, element and quantifier. Let us take an example. "Some men are clever." Set-theoretical analysis presupposes that the sets "men" and "clever person" must be specified. It is not easy. We have to answer such questions: Is the Neanderthal man a man? Is it possible to include into that set future persons, e.g., in CLX century? Will mankind exist in that time? In practice we don't know such things. Actually in spite of the logicians' drill we don't separate the set "men" as a subject from "some" as a quantifier. We think that the subject of our thought is "some men" or indefinite man – a man. From our point of view "a man", "the man" and "any man" are three kinds of "man". All of them can be a subject of the corresponding judgments.

In general we can denote the definite object (thing, entity), i.e., the object by the symbol t, an indefinite object, i.e., an object by the symbol a, and an arbitrary object – any object by the symbol A. Formulae t, a, A are elementary well-formed formulae (WFF) of our formalism – Ternary Description Language.

The other types of WFF are formed in following manner:

II {A)A – Arbitrary thing (= object = entity) has arbitrary property. Certainly, that judgment is wrong. But it is a WFF. We can substitute instead of A, any WFF. All the results of such substitutions must be WFF. e.g., (A)a, (t)a, (a)t are the results of the substitutions of A by elementary formulae. In the case of ((A)a)a, we have a substitution non-elementary formula (A)a instead of A (thing) and elementary formula a instead of A (property).

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3. A(A) – Arbitrary thing has arbitrary relation. WFF: a(A), a(t), t{a), a(a(A)) are special cases of the WFF of that type.

4. (A*)A. This type of WFF differs from (II) in the direction of the predicate relation. That formula means that an arbitrary property belongs to an arbitrary thing. Of course it isn't true. But some special cases of (IV) are true: (a*)A, (a*)t, (a*)a, (a*)(a)t.

V A(*A). This formula is analogous to the preceding one. It means that an arbitrary relation can be realized on an arbitrary thing. It is wrong. The next formulae of that type are true: A(*a), t(*a), a(*a), a(t)(*a).

The formulae of the (II) – (III) types may be called direct ones and the formulae of the (IV)-(V) types – inverse ones.

VI [A]. A formula of this type means that which may be called the conceptual closure of the formula A. If A is of a propositional nature, i.e., express a proposition, then [A] denotes the concept corresponding to the statement A.

The conceptual closure of (A)A gives us the formula [(A)A] which must be interpreted as an "arbitrary thing possessing arbitrary property", in particular [(a)a] – a thing possessing a property. Correspondingly [(A*)A] – arbitrary property inherent in an arbitrary thing. [{a*)a] – a property inherent in a thing. By the same token, mutatis mutandis, we can interpret the formulae [A(A)], [A(*A)].

We may say that the formulae of type (II)-(V) are open, while the formulae with square outermost brackets are closed. If the formula A is closed, its closure doesn't change that formula. Therefore [[A]] means the same as [A].

VII {A}. Curly brackets have an ancillary character. They are used in the case when an inclusion of one formula into another as a subformula leads to ambiguity. For example, the formula (A)a(A) is well-formed. But it is possible to understand it in two ways: as "A possesses the property a(A)" and as "A possesses the relation (A)a." It is quite possible to adopt both interpretations simultaneously. But when only one of them should be adopted, the corresponding subformula is enclosed in curly brackets: (A){a(A)} or {(A)a}(A).

VIII A,A This type of WFF is a simple list of WFF. We shall call the formulae of such a type free lists because it doesn't suppose any relation between its components.

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Note that the order of the formulae in a list is not simply inessential; it is ignored, just as the differences between typographical marks for one and the same symbol are ignored. For the reason given, the combination of symbols t, a and a, t are regarded as one and the same combination of symbols.

Nevertheless the order of symbols is very essential in the other types of WFF. We saw it above on the examples of direct and inverse formulae. The importance of that order has its manifestation in the role of a symbol's place in a formula in the interpretation of that symbol. It doesn't hold true for the symbol of definite object – t, because t must be defined in advance without regard to a formula in which t is included. But the concrete meaning of an arbitrary object A and especially – indefinite object – a are function of their surroundings in a formula.

Let us distinguish the first – initial, and the second – final, parts in every formulae listed above. In direct formulae initial parts are included in parentheses. They denote things. Those are true both for the open and closed formulae. If the initial parts of the formulae are underlined we receive: {(A)A}, {A(A)}, [(A)A], [A(A)]. In inversed formulae the initial parts denote properties and relations. Accordingly we have: {{A*)A}, {A(*A)}, [{A*)A], [A(*A)].

Note that a marked out formula is not a formula of a new kind. Marks are only means for the better understanding of a formula. In the open formulae: {(A)A), {A(A)}, {(A*)A}, {A(*A)} both A are completely arbitrary objects. Nevertheless in the closed formulae: [(A)A], [A(A)], [(A*)A)] and [A(*A)] the symbols of the completely arbitrary objects are placed only on the final parts. The arbitrariness of A on the initial parts of the formula are restricted. In [(A) A] the symbol A on the initial part of the formula denotes the arbitrary thing, which is having an arbitrary property. Of course, we cannot find such object in our world. But [{A)t] – an arbitrary thing having the property t, is a real object, if t is real. Correspondingly [t(A)], [(t*)A], [A(*t)] are examples of other kinds of restricted arbitrary things.

An indefinite thing, the symbol of which is placed on the initial part of an open formula, has an unlimited range of indefiniteness. e.g., it is so in the formulae {(a)a}, {a(a)}, {(a*)a}, {a(*a)}. But the second a – in the final parts of those formulae – has a restricted indefiniteness. In (a)a it is such an a which is a property of the first a. Correspondingly in {a(a)} the second a is a relation of the first a. In {(a*)a} the final a

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is a thing to which an initial a is prescribed as a property, and in {a(*a)} – as a relation.

In the case of complicated formulae which consist of many sub-formulae of various types we can find the initial part, i.e., the beginning of the formula step by step, e.g., in the formula ([((A*)A)A])A we can determine the initial part – [((A*)A)A]. The next step is finding the initial part of that initial part. It is (A*)A. And finally we received the beginning formula at whole – (A*)A. The indefiniteness placed on the beginning of an open formula will be called initial, while the indefiniteness restricted by the sentence's context – contextual.

In the closed formulae the indefiniteness can be contextual even on the initial place, e.g. {{a)t]. But in the case of [(a)a], there is no actual limitation, because a thing always has a property.