THE SUBSIDIARY CONCEPTS OF THE
TERNARY DESCRIPTION LANGUAGE.
The most fundamental relations in a logical calculus are implications. They are usually defined with the aid of the truth function tables as a relationships between statements. Our position differs from the traditional one in two aspects. First of all, we are not assuming truth-falsity as primitive concepts. Analogues of truth and falsity will be defined below as
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derived concepts. Further, we recognize in our system the equivalence of sentences and concepts in principle (expressed by open and closed formulae respectively). Both sentences and concepts can act as antecedents and consequents of implications.
We shall make use of several types of implications. The first, called attributive, is defined with the aid of the following formula:
{ i A Þ ii A } =def ji A j [(a)ii A] (3.1)
Here, the definiens* expresses identification of the object denoted by i A with some object possessing properties expressed by ii A. This holds in categorical sentences expressed with the aid of the copulative verb “is”. When we say that a square is a rectangle, we mean that a square is identical to some object endowed with the properties of a rectangle.
The sequence of objects i A and ii A expressed as a formula of the list type i A,ii A can be composed with various aims. If something is known about the objects i A and ii A, then both these objects could be joined by a single pair of curly brackets merely for the sake of simplification. The list i A,ii A will mean the same thing as i A and ii A separately. Instead of writing a(*i A), a(*ii A), we may write
a(*{i A, ii A}). In this connection it is necessary to note, that here a relation a is not considered as a relation between objects i A and ii A, but only as a relation in (or to) objects i A and ii A separately.
If objects of the list are considered separately, not relating to each other, then, even without denying the presence of a relation between them, we are dealing with an ordinary, simple free list.
If the list somehow relates objects to each other, then, in this case we shall call it related and denote it by {i A · ii A }. We can give the following definition of such a list:
{ i A · ii A } =def [ (iii A){[ A(*i A, ii A)] Þ [a(*iii A)]} ] (3.2)
The meaning of this definition resides in the fact that for a related list
i A · ii A there is such an object iii A that any relation to an object i A or ii A will simultaneously be a relation to the object iii A. Consequently, through a certain intermediate object – iii A, the objects i A and ii A turn out to be related to each other.
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Using the concept of a related list, let us define a new type of implication, which we shall denote by the symbol É :
{ i A É ii A } =def ji A j{ii A · a} (3.3)
The implication defined by this formula is called mereological, since it corresponds to a generalized understanding of the relationship between a part and a whole, according to which a part may also be a subset and an element of a set. That relationship was studied by St.Lesniewski in his “Mereology” (Luschei, 1962).
The definiens of definition (3.3) means that a whole – i A is identical to a part – ii A, to which something has been added. For example, Ukraine mereologically implies Odessa, because Ukraine is equal to Odessa with something else added.
The third type of implication is relational. It is analogous to the attributive implication, but the role of attribute in it is played by a relation. Let us denote the relational type of implication by the symbol f. We shall define it with the help of the next formula:
{ i A f ii A } =def ji A j[ii A(a)] (3.4)
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* “Every definitional formula has two components, one containing the new sign and the other not. The component containing the new sign is called the definiendum; we shall follow the practice of writing the definiendum as the first, or left, component of the definition. The other component of a definition contains only earlier signs: it is called the definiens.” (R.Carnap, Introduction to Symbolic Logic and its Applications, Dover Publications, New York, 1958, p.64.) In the formula (3.1) the definiendum is {i A Þ ii A } and the definiens is ji A j[(a)ii A].
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E.g. a map of a city (i A) implies relations between streets of that city (ii A). An object i A is identical to some object with relations ii A.
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We can generalize the three types of implications defined above in the concept of neutral implication. We use a simple arrow to denote it:
{ i A ® ii A } =def ([ A(*i A · ii A)]){[(i A Þii A*)iii A],
[(i A Éii A*)iii A], [(i A fii A*)iii A]} (3.5)
According to this definition, neutral implication is a relation between a list of related objects, which possesses an arbitrary property that can be ascribed to the attributive, mereological, and relational implications. For example, if it is shown that the attributive, mereological, and relational implications all possess the property of transitivity, then this property may be extended to the neutral implication. But if some property is inherent only to the attributive or only to the mereological or relational implication, then this is not sufficient to ascribe it to the neutral implication. At the same time, if a property is inherent to the neutral implication, then it is inherent to the attributive, mereological, and relational implications as well.
Note that in defining implications we did not make any use of valent values – Truth and Falsity. This grants us the possibility for inverting the problem, i.e. for posing the question of defining valent values through an implication introduced independently of them.
Truth and Falsity are regarded as some properties of TDL formulae, not only for open formulae but closed as well. In its turn, they have properties themselves. Truth has such properties as: a) It “carries over” from the antecedent of an implication (neutral, and hence, also every) to its consequent; b) It is preserved by repetition – truth of a true formula denotes a true formula.
Falsity has properties of the opposite nature: a) It “carries over” from the consequent of an implication to its antecedent; b) It is not preserved by repetition – falsity of a false formula denotes a true formula.
In as much as we define all those properties formally, we will obtain the formal definitions of a true and false formulae.
Let us denote the property of “carry over” from the antecedent of an implication to its consequent as a c . Here the index c is from the word “consequent”. The formal definition of a c will be:
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ac =def [(i a){ { ii A ® iii A }® {[(ii A)i a] ® [(iii A)i a]} }] (3.6)
Analogously, the opposite property of “carrying over” from the consequent to the antecedent – a a (index a is from the word “antecedent”) will have the definition:
aa =def [(i a){ { ii A ® iii A }® {[(iii A)i a] ® [(ii A)i a]} }] (3.7)
If a c preserves itself by repetition, we denote it as a cp (index p is from the word “preserve’) and express it formally in the following way:
acp =def [(i ac){ {([(ii A)i ac]) i ac } Þ {(ii A)i ac} }] (3.8)
In the opposite case we use the denotation a an (index n is from “non”) and give its formal definition in such a way:
aan =def [(i aa){ {([(ii A)i aa]) i aa } Þ {(ii A) ac} }] (3.9)
And finally we can express the property of returning of a cp through the repetition of a a. Denote it by a cpr (index r is from “returning”):
acpr =def [(i acp){ {([(iii A)ii aa]) ii aa } Þ {(iii A)i acp} }] (3.10)
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Now we can define the “true” and “false” formulae:
(i A)T =def (i A) a cpr (3.11)
(i A)F =def (i A) a an (3.12)
Of course we do not pretend to have given definitions of Truth and Falsity in their broad philosophical connotations. We shall speak only of the construction of their formal, logical models.
Let us pay attention to the structure of the definitions (3.11-12) as well as all the previous definitions. They are built with the aid of A supplied with the iota operators. This gives us the possibility to formulate the next rule of definitional deduction:
If different occurrences of A in a definition are bounded by the same iota operator, then instead of all these occurrences one may substitute an arbitrary (but anywhere the same) formula. The iota operator may be preserved or omitted provided that single iota operators do not appear.
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Thus our definitions can be transformed into an infinite number of various definitions. So, our definitions we can call definition schemes.
For example, let us substitute both occurrences of A in (3.11) for the formula (A)a. Instead of (3.11) we will obtain:
(i{(A)a})T =def (i{(A)a}) a cpr (3.13)
In the case of omitting of the iota operator we will have:
( (A)a )T =def ( (A)a ) a cpr (3.14)
A subformula i A in a formula (i A)T can, in its turn, be valent, i.e. ends with T or F or with other valent sign, which can be defined at a later time. Continuing further, we shall eventually arrive at a subformula which contains no valent signs. This will be the formula’s nucleus. The collection of all valency signs which characterizes the nucleus, may be called the valency ending. If a formula is identical with its nucleus, i.e. has no valency ending, it may be called nucleary. Such is the subformula (A)a in the makeup of the formula ((A)a)T. Formulae may also be without any nucleus, i.e. nucleariless, e.g. ( (T)a )T or ( (T)T)T.
Nucleary formulae will be included in the class of the not-valent formulae. Not-valent formula may be used in two ways: as a nucleus of valent formula or as an independent one. Note that a not-valent formula, while not having valency, has some meaning, which is expressed by its structure. For example, the formula (a)A means that a thing has any property. It is true that the given formula has precisely this meaning. As not valent we can look at this formula for instance putting the question: is it a well-formed formula?
A not-valent formula can be defined with the help of another not-valent formula. In this case the definition expresses the identity of meanings of the two correlated formulae. The valent formulae were defined through not valent formulae in the definitions (3.11-12). It means that valent formulae are special cases of not-valent ones.
But, not all not-valent formulae are equivalent to some valent ones. Because of this we do not accept the so-called assertion principle, according to which the utterance of a statement is an assertion of its truthfulness (Carnap, 1946). Therefore, our notation for any not-valent formula, e.g. t or (t)a, does not mean that a formula, which has been written down, lays a claim to truth. t assumes truth if this formula will be the nucleus of the formula (t)T. Correspondingly, the truth of (t)a is assumed if there is ((t)a)T. The endings T or F can be used twice, three times, et seq. For example, it is possible to have ((((t)a)T)T)T or ((((t)a)F)F)F. The first formula is equivalent to ((t)a)T, the second – to ((t)a)F.
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There can be mixed endings containing different valency signs, for example ((A)F)T. The given formula is interpreted as follows: the assertion that “an arbitrary object is false” is true.
This should stir up protest on our part for the thought expressed here, that the entire world is an illusion. But how shall we express this protest? Apparently, we may write F in place of T, i.e. introduce a second negation. Hence ((A)F)F. However, in accordance with the definition of (A)F, we shall get that the repeated falsity returns to the object’s truth. Therefore, while (A)F signified the falsity of an arbitrary object, ((A)F)F will denote the truth of an arbitrary object, i.e. (A)T. Thus we have a recognition of the truth not only of the Pacific Ocean or of New York city but also of the present King of France, the naked King’s clothes, the round square etc, because all these objects can substitute A in (A)T.
What we have said indicates that falsity, in the form defined above, is not a fully adequate model of negation. Nevertheless, the more adequate model of it can be obtained on the basis of the valent formulae already defined.
Let us call the falsity, which we defined above, contrary. It can be applied directly to the nucleary formulae. Contradictory falsity (contradictory negation), which we will denote by the symbol n, can be applied only to the formulae which have already been valent. It can be defined with the help of the following formal definitions:
((A)T)n =def (a)F (3.15)
((A)F)n =def (a)T (3.16)
( ((A)i A)T )n =def ( (a)i A )F (3.17)
( (i A (A))T )n =def (i A (a) )F (3.18)
( ((i A*) A)T )n =def ( (i A*)a )F (3.19)
( (A (*i A))T )n =def (a (*i A) )F (3.20)
Instead of definitions (3.17-20) we may use only one expression:
( (A R i A)T )n =def (a R i A)F (3.21)
Here R denotes the structure of a formula from the following list:
(A)i A, i A(A), (i A*)A, A(*i A), [(A)i A], [i A(A)], [(i A*)A], [A(*i A)].
This structure is the same on both sides of the definition scheme (3.21).
The usage of the definition scheme (3.21) allows us to add some more definitions to the definitions (3.17-20):
(( [(A)i A] )T )n =def ( [(a)i A])F (3.22)
(( [i A(A)] )T )n =def ( [i A(a)])F (3.23)
(( [(i A*)A] )T )n =def ([(i A*)a])F (3.24)
(( [A(*i A)] )T )n =def ([a(*i A)])F (3.25)
The scheme (3.21) can be generalized in the following formula:
(( ii A Q {A R i A} )T )n =def (ii A Q {a R i A})F (3.26)
Analogously we should have the next scheme:
(( ii A Q {A R i A} )F )n =def (ii A Q {a R i A})T (3.27)
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Here Q denotes the structure of a formula from the list of 16 formulae:
(A)i A , i A(A) , (i A*)A , A(*i A) , [(A)i A] , [i A(A)] , [(i A*)A] , [A(*i A)] ,
(i A)A , A(i A) , (A*)i A , i A(*A) , [(i A)A] , [A(i A)] , [(A*)i A] , [i A(*A)].
As an example of the application of the definition schemes given above, let us consider the formula ( ( A Þ t )T )n . Using the definition of the attributive implication, we obtain: (( jA j[ (a)t ] )T )n . The next step should be the implementation of j-identity definition: jA j[ (a)t ] =def (A)[( [(A)[ ([ (a)t ]*)A ] ]*)A]. Let us now substitute i A in the formula (3.17) for [(A) [( [(A)[ ([ (a)t ]*)A ] ]*)A] ] (iota operator in i A can be omitted according to the rule of definitional deduction). We obtain:
( ((A) [(A) [( [(A)[ ([ (a)t ]*)A ] ]*)A] ] )T )n =def
( (a) [(A) [( [(A)[ ([ (a)t ]*)A ] ]*)A] ])F
And, returning to abbreviations: ( ( A Þ t )T )n =def ( a Þ t )F.
The contradictory negation can be applied in the opposite direction – from a to A. For this case, we have the following definitions:
((a)T)n =def (A)F (3.28)
((a)F)n =def (A)T (3.29)
( (a R i A)T )n =def (A R i A )F (3.30)
( (i A R a)T )n =def (i A R A )F (3.31)
((a R i A)F )n =def ( A R i A )T (3.32)
( (i A R a)F )n =def (i A R A )T (3.33)
We can generalize all these definitions in two schemes:
(( ii A Q {a R i A} )T )n =def (ii A Q {A R i A})F (3.34)
(( ii A Q {a R i A} )F )n =def (ii A Q {A R i A})T (3.35)
Returning to our initial example, we can easily express our philosophical attitude with the aid of contradictory falsity. Instead of ( (A)F )F we must write ( (A)F )n.
Let us highlight one important point. The definitions and definition schemes given above show that the contradictory negation, as distinct from F, is not implemented in any formula. The sphere of the application of the definitions and schemes that contain n is restricted to the specific types of formulae, namely those which contain A or a without iota operators as subformulae.
Taking into account the contradictory falsity, we obtain the three types of valent endings of TDL formulae: T, F and n. Except that it is possible to introduce also the fourth valency, when a formula has the valencies T and F simultaneously. For example, it is true that (a)T, and also true that (a)F. In general form it can be expressed as: (a){T,F}. Formulae which have the values T and F simultaneously may be called ambivalent ones. It is possible to combine the ambivalence and contradictory negation. For example, if there is ((A)T)n and ((A)F)n simultaneously, we can express it in the united formula: ((A){T,F})n.
Valencies, which were considered above, may be called definite ones. Indefinite valency may be ascribed to non-valent formulae, if they can have valency in principle. In addition to this it can be defined quasidefinite valency, when one valency T or F is known and other is possible. A quasitrue formula is true or ambivalent one. Let denote such a valency by {T, }. Correspondingly quasifalsity is denoted as {F, }. Quasitruth does not contradict to truth, and quasifalsity does not contradict to falsity of the corresponding formulae.
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Let us return to the assertion principle. It can be expressed in such a form:
(i A Þ (i A)T )T (3.36)
We contrarily negate it. Therefore there is:
( (i A Þ (i A)T )T )F (3.37)
Note that the application of the assertion principle in classical logic has two limitations. First, it is applicable only to statements, which we express by the open formulae. In classical logic only statements can be evaluated as true ones. Second, it is applied only to a formula as a whole, not to separate parts of it, i.e. subformulae. In the statement calculus the utterance a ® b may be regarded as true, but its components a and b are not regarded so. Their valency is unknown. If we know that a is true, we must write it separately: a ® b , a .
In our system closed formulae can be valent just as open ones. Similarly, subformulae can be valent or not. For example, in (3.36) the subformula i A, which denotes the consequent, is valent, just as the formula of implication in its entirety. But i A in the antecedent may be not valent.
The abandonment of the assertion principle leads to the usage of a large number of valency signs, which would make formulae too unwieldy. In order to avoid it, let us consider conventions for simplifications:
a) We will consider as not-valent such a formula, where the valency is unknown. Valency markers are determined on the basis of axioms and rules of inference, so these markers may be omitted in the case when their determinations are trivial. For example, in the formula (t,a,a,t)T there is no necessity to mark valencies of the every subformulae. According to the rule for lists RC1 all of them are true.
b) If the valency of the whole formula does not depend on valencies of subformulae, those subformulae may be regarded as not-valent. E.g., the formula
( (a){T,F} )[( (A){T,F} )n] (3.38)
has the valency F regardless of subformulae valencies. Therefore we may have the simpler formula (a)A instead of (3.38).
c) It is supposed that the definiens and definiendum are true simultaneously. Therefore if the definiendum is true, the definiens is true also. In this case the valent sign T may be omitted both after the definiendum and definiens.
d) Because in this paper implications of any kind will be seldom used as not-valent formulae, and on the contrary, true implications will be used very often, it is convenient to agree here that the absence of valency sign after an implication denotes the mark T. If an implication lacks the valency T, we will always follow it with one of these symbols: F, T)n , F)n , {T,F} , {T, } , {F, } . If an implication must be considered as a not-valent formula, it will be denoted by inverted commas, e.g. “A Þ a”.
To formalize the system parameters values we need to define some specific “objects” by application of previously defined operations to the basic objects a, A, t of TDL.
Let t¢ be an indefinite object which is different from t. Formally:
t¢ =def [(i a){ ( { i a É t } · { t É i a } )F }] (3.39)
In accordance with the accepted notations, the definiens of (3.39) means that it is impossible to have both implications simultaneously. And this impossibility is the property of i a.
An indefinite subobject – tÈ – has a definition:
tÈ =def [(i t¢){ t É i t¢ }] (3.40)
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That is some object, different from t, possesses the property that t contains it. In some cases such an object corresponds to the notion of a physical or extensional part of the object t. E.g., a man contains his head as a physical part. A different kind of a part is an intensional one. In this case it may be a collection of properties. A man “contains” an animal in the sense that he possesses properties of an animal, so the collection of these properties is a part of a man. This part may be called qualitative or attributive one. An intensional part may represent also some collection of relations in the object t. In this case we may call it relational one. E.g., a city contains streets as relations between buildings. Buildings themselves are extensional parts of a city.
An indefinite superobject – tD – is defined as:
tD =def [(i t¢){ i t¢ É t}] (3.41)
Unlike the preceding (3.40) here the question is not about t containing an indefinite object i t¢ , but of t being contained in i t¢. In extensional sense t¢ may be e.g. a family, in which a man t is included. In intensional sense t¢ may be t in some special state, e.g. t¢ is a young man, where t is simply a man. It is an attributive superobject. The object t, to which a some relation is added, is relational superobject. E.g. Romeo and Julia in love with one another construct such a relational superobject different to that couple without love.
An indefinite disparate – t° – is defined as an object, different from t, which possesses two properties simultaneously: it is neither a subobject nor a superobject. Formally:
t° =def [(i t¢){ {((i t¢ *) tÈ)F} · {((i t¢ *) tD)F} }] (3.42)
Finally, we will give the definition of the limited definite object. The idea of such an object is the following. The addition of something to a definite object does not usually mean that the entity has become different. For example, if we put gloves on the cat in boots it will still remain the cat in boots, in spite of the fact that gloves are not a part of boots. The situation is different if we have a cat in boots only, as should be supposed according to Perro’s well-known tale. In order to emphasize this point, we should say that every addition that preserves a definite entity t (i.e. {t ·i A} Þ t ) must occur in it (i.e. tÉ i A ). The latter is presupposed by the former and is an element of the former. A definite object possessing the indicated property we denote as Lt and call it “the limited object”.
Lt =def [( t ){ { { t ·i A } Þ t } É { tÉ i A} }] (3.43)
All the definitions that were given above may mutatis mutandis be applied to the iotafied object:
i A¢ =def [(ii a){ ( { ii a É i A } · { i A É ii a } )F }] (3.44)
i AÈ =def [(i A¢ ){ i A É i A¢ }] (3.45)
i AD =def [(i A¢){ i A¢ É i A}] (3.46)
i A° =def [(i A¢){ {((i A¢ *) i AÈ )F} · {((i A¢ *) i AD )F} }] (3.47)
L i A =def [(i A ){ { { i A ·ii A } Þ i A } É { i AÉ ii A} }] (3.48)
Let us finish this section of our article with the two definitions. First is the definition of two kinds of disjunction:
{i A Ú ii A } =def { {(i A)F ® (ii A)T }·{(ii A)F ® (i A)T } } (3.49)
{i A W ii A } =def { {i A Ú ii A } · {( (i A)T · (ii A)T )F} } (3.50)
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Second is the definition of interjective relation. It is expressed with the word “inter” (between). A relation exists between components of a related list, if it exists in that related list, but does not exist in components of that list taken separately. E.g., the relation “husband” exists in a couple (Peter, Mary), but it does not exist in Peter and in Mary taken separately. Formally interjective relation may be expressed in the following definition:
i A inter(ii A·iii A) =def { (i A(ii A·iii A))T }·{(i A(ii A,iii A))F} (3.51)
The index inter may be reduced to int and even to i .
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