S~AS~AN K. SAVMJAN - P. A. SO~Or~VA 
FORMAL METALANGUAGE AND FORMAL THEORY AS 
TWO ASPECTS OF GENERATIVE GRAMMAR 
As is known, metalanguage is a language by means of which an- 
other language is described. The latter is called object-language. 
One and the same language may play the role of object-language 
and metalanguage. For example, the Russian language taken as an 
object of linguistic description is an object-language, the Russian lan- 
guage used for a linguistic description of Russian is a metalanguage. 
There are non-formal and formal metalanguages. For example, 
the Russian language used in ordinary grammars of Russian as a means 
of describing the Russian language is a non-formal metalanguage. 
Formal metalanguage is an artificial language, defined by deductive 
rules of construction, and used to describe natural languages. 
The problem of formal metalanguages for linguistic descriptions 
is a broad topic which we do not mean to-exhaust. We shall restrict 
ourselves to clarifying the role of formal metalanguages in constructing 
generative grammars. 
Every generative grammar is a formal theory. For example, a gen- 
erative grammar of Russian is a formal theory of Russian, a generative 
grammar of English is a formal theory of English. We shall focus our 
attention on the question of correlation between formal metalanguages 
and formal linguistic theories understood as generative grammars. Let 
us take a concrete example: formal metalanguage and formal theory 
as two aspects of applicative generative grammar. This concrete exam- 
ple will make it possible, we hope, to draw certain general conclusions 
about the role of formal metalanguages in generative grammar. 
We shall start with formal metalanguage in applicative generative 
grammar. We call this metalanguage a universal operator language. 
The universal operator language is defined by the grammar which 
is an ordered quadruple 
(v,, V ,F, R5 
:/jr 
64 SEBASTIAN K. ~AUMJAN- P. A. SOBOLEVA 
where: 
1. V, is the lexicon of symbols whose denotata belong to a finite 
set of elementary categories called elementary episemions. The elemen- 
tary episemions are denoted by the symbols 0c and \[3. The first is in- 
terpreted as the category of terms, and the second - as the category 
of sentences. The episemion \[3 is called the distinguished episemion. 
Complex categories called episemions are constructed from elemen- 
tary episemions. The construction of episemions is effected with the 
help of the function A, which maps episemions into each other. The 
rules for constructing episemions are the following: 
(a) elementary episemions are episemions 
(b) if p and q are episemions, then A pq is also an episemion. 
Rule (b) presented as a tree looks as follows: 
(1) P q 
Pq 
For example: 
(2) A~I3 
The episemion A0~ may be interpreted as the category of one-place 
predicates (i.e. the function which maps terms into sentences), for exam- 
ple burn, arrive, sleep, stay etc. 
(3) 0~ 0c 
A0c~ 
The episemion A0~0~ may be interpreted as the category of term identi- 
tiers (i.e. the function which maps terms into terms), for example 
big, good, John's, burning (in burning eyes, burning house) etc. 
(4) ~ A0c\[3 
A~A~\[3 
The episemion A0cA~\[3 may be interpreted as the category of two-place 
predicates (i.e. the function which maps terms into one-place predi- 
cates or, in other words, the function which maps two terms into sen- 
tences), for example read, see, expect, like etc. 
-% 
FORMAL METALANGUAGE AND FORMAL TItEORY 65 
(s) 
The episemion AA~130c may be interpreted as the category of depred- 
icators (i.e. the function which maps one-place predicates into terms), 
for example -al (in'arrival), -ation (in relaxation) etc. 
(6) ~ Aoc~ 
The episemion A~A,~ may be interpreted as the category of denomi- 
native adjectivisers (i.e. the function which maps terms into term 
identifiers), for example -y (in stony path), -less (in heartless man), 's 
(in John's house), of (in the house of John) etc. 
(7) a~ A~ 
The episemion AA0~A~ may be interpreted as the category of adjec- 
tivising depredicators (i.e. the function which maps one-place predi- 
cates into term identifiers), for example -ing (in burning house), -7 (sleepy 
child) etc. 
2. V s is the lexicon of symbols whose denotata are elementary 
meanings. These symbols are called elementary semions. For example 
T - an elementary semion which is interpreted as an object belong- 
ing to the category of terms. 
P1 - an elementary semion interpreted as an object belonging to 
the category of one-place predicates. 
P~ - an elementary semion interpreted as an object belonging to 
the category of two-place predicates. 
A - an elementary semion interpreted as an object belonging to 
the category of term identifiers. 
D r - an elementary semion interpreted as an object belonging to 
the category of depredicators. 
Da - an elementary semion interpreted as an object belonging to 
the category of adjectivizing depredicators. 
+, 
66 SEBASTIAN K. ~AUMJAN- P. A. SOBOLEVA 
Sfa - an elementary semion interpreted as an object belonging to 
the category of denominative adjectivisers etc. 
3. F is a so-called assignment function which places each elementary 
semion in correspondence with one or several episemions. The work 
of the assignment function may be represented by the formula 
eX 
where e is any episemion and X is a semion with which it has been 
placed in correspondence, for example 
0~T 
A~ P1 
A~A~ P2 
Aoco~ A 
Aoc~o~ D r 
AA~SA~ DA 
A~A~ Sfa etc. 
4. R - are rules for constructing combinations of elementary se- 
mions which are called semions. These rules are as follows: 
(a) an elementary semion is a semion; 
(b) if X is a semion which belongs to the episemion A pq, and 
Y is a semion, belonging to the episemion p, then (XY) is a semion, 
belonging to the episemion q. 
Rule (b) presented as a tree looks as follows: 
(8) pY q(xr) 
We shall call X an operator, Y - an operand, and (XY) is the re- 
sult of application of X to Y. P,.ule (b) is called the rule of application 
of semions. 
Ikule (b) can be illustrated in the following way: 
(9) A~Px 0cT 
(P1 T) 
If P1 is interpreted as stay and T as John then the semion P1 T is in- 
terpreted as the sentence John stays. (Note that in the abstract operator 
language the operator always precedes the operand. So a natural lan- 
FORMAL NLETALANGUAGE AND FORMAL THEORY 67 
guage analogue of the semion/2, T is stays John rather than John stays). 
Below are given several more illustrations of rule (b): 
A~0c~10~,Da A0C~Px 
(10) Aococ (DAP~) oct 
oc((DAP~)T) 
If the elementary semion DA is interpreted as -ing, Pz as burn (intr.), 
then the semion DAP1 is interpreted as burning. If T is interpreted as 
house, then the semion ((DAP1)T) is interpreted as burning house. (The 
exact order of elements being ((-ing burn) house)). 
AocAococSf~ ocT Aoc~ocD T Aoc~P1 
(11) Aococ(SfAT) oc(DrP~) 
oc ( (SL T) (DrP~)) 
If Sfa is interpreted as 's, T - as John, then Sf,~ T corresponds to John's. 
If D r is interpreted as -al, Px - as arrive, then DrPI corresponds to ar- 
rival. The derived semion ((Sf.4 T) (DTP))serves as a genotype analogue 
of the noun phrase John's arrival (the exact order of elements being 
(('s John) (-al arrive))). 
Such is the grammar which defines the construction of the univer- 
sal operator language. If an empirical interpretation is assigned to epi- 
semions and semions, the universal operator language may serve as 
a formal metalanguage for the description of any natural language. 
The universal operator language has sufficient potentialities to construct 
semions which serve as abstract analogues of sentences of any degree 
of complexity. By way of an example we shall construct semions 
which serve as abstract analogues of the following complex phrases: 
I see a burning house (12) and 
I expect John's arrival (13). 
The tree below shows the construction of the genotype analogue 
of the first phrase: 
A~ (DAp~) ~ T 1 
(12) AocAoc~P~ oc ( (DAP1) T ~) 
Aoc~(P~((D~Pz)Tz)) ocT 2 
~((P~((DaPI)T~))T 2) 
68 SEBASTIAN K. SAUMJAN- P. A. SOBOLEVA 
The final semion ((P2 ((DaP1) 7"i)) T ~) which is an object belonging 
to the category of\[3, i.e. sentence, is an abstract analogue of a two-place 
predicate sentence with a participle defining one of its terms. 
The construction of the second semion is shown below: 
A~A~Sfa ~ T 1 AA~f3o~Dr A~3P1 
Ao~o~(SfA T ~) o~(DrP~) 
(13) T1)(DTP )) 
A~(Pz((SfAT~)(DrPI)) ) ~W 2 
~ (P, ( ( Sfa T~) (DrP1) ) T *) 
The final semion P,((SfAT' ) (DrP,))T ~ which also belongs to the 
category of \[~, i.e. sentence, is an abstract analogue of a two-place pred- 
icate sentence with a nominalized phrase as one of its arguments. 
However complex the constructed semion may be, the process of 
construction fails to show how two primitive structures of the type 
I see a house and The house is burning are transformed into (12), the 
second becoming a participial phrase in (12), or how two primitive 
structures of the type I expect smth. and John is arriving are transformed 
into (13), the second becoming a nominalized phrase in (13). 
Though the construction of semions accounts for the generation 
of the abstract analogues of sentences of any degree of complexity it 
does not aim to show such essential grammatical processes as nomi- 
nalization or adjectivation of phrases, formation of complex predicates 
from simple ones or of simple sentences from complex sentences etc. 
So in spite of the abstract character of the universal operator language 
it does not allow us to rise above the level of taxonomic description 
of natural languages. 
The necessity for the formal theory of natural languages follows 
from the two-level principle in linguistic studies. According to the 
two-level principle every natural language is stratified into two lan- 
guages: the phenotype language and the genotype language. The 
phenotype language is a natural language as is given in immediate 
observation. The genotype language is hidden, not given in immediate 
observation; it is a construct language which consists of two sub-lan= 
guages: the basic genotype language and the derived genotype lan- 
guage. The basic genotype language is the language of thought rep- 
resentation; the expressions of the basic genotype language which 
consist of elementary predicates may be identified with the thoughts 
which are to be embodied into a linguistic form. The embodiment 
FORMAL METALANGUAGE AND FORMAL THEORY 69 
of thoughts into a lingustic form is effected in two stages. At the first 
stage the expressions of the basic genotype language, which are identi- 
fied with thoughts, are transformed into the expressions of the derived 
genotype language, regarded as abstract linguistic forms which em- 
body thoughts. At the second stage the expressions of the derived 
genotype language are transformed into the expressions of the phe- 
notype language regarded as the concrete linguistic forms of thought 
embodiment. Thus, the derived genotype language may be considered 
as an intermediary language between the basic genotype language and 
the phenotype language. 
The transformations used to convert phrases of the basis genotype 
language into phrases of the derived genotype language may be describ- 
ed with the help of logical operators called combinators. I 
Since natural languages serve the purpose of communication be- 
tween people, the grammar of every natural languages must possess 
the means for the transformation of thoughts into linguistic forms 
and, vice-versa, the means for the transformation Of linguistic forms 
into thoughts. 
Grammar viewed in such a way is a hypothetic transducer, which 
is not given in immediate observation, but which makes it possible to 
transform the phrases of the basic genotype languages into the phrases 
of the derived genotype language, and the phrases of the derived gen- 
otype language into the phrases of the phenotype language, and vice- 
versa, the phrases of the phenotype language into the phrases of the 
derived genotype language, as well as the phrases of the derived geno- 
type language into the phrases of the basic genotype language. 
The formal theory of natural languages must use the universal 
operator language as a formal metalanguage for a description of the 
basic genotype language, the derived genotype language and the phe- 
notype language. 
Below we shall specify the type of rules for .postulating the basic 
genotype language and the type of rules for obtaining the derived gen- 
otype language. The basic genotype language consists of one-, two- 
and three-place predicates obtained through the rule: 
(14) S ~ I CJ P3 T~gT* 1 
x See. S. K. ~AUMJAN (1971), pp. 127-133. 
70 SEBASTIAN K. SAUMJAN- P. A. SOBOLEVA 
In (14) S is a global symbol of sentence, i.e. a semion belonging to 
the category ~. /91 in (14) is a one-place predicate with a sentence as 
its argument. Its episemion is A~. P2 is a two-place predicate with 
a sentence and a term as its arguments. Its episemion is A~A.0~. P3 is 
a three-place predicate with a sentence and two terms as its arguments. 
The episemion ascribed to it is A0~/X~A~. Cj is a two-place operator 
of conjunction with sentences as its arguments. Its episemion is A~A\[5~. 
is a sentence which is an argument of P1, P2 and P3 (an embedded 
sentence). Its episemion is ~. The status of T has been discussed on 
page 65. It is an object belonging to the category of terms. In (14) it 
is an argument of P1, P~ and P3~ ~' 
The one-place predicate P1 corresponds to modal words in nat- 
ural languages, such as possibly, probably, perhaps etc. It may also cor- 
respond to aspectual words, such as begin, continue, stop. The two-place 
predicate P2 corresponds to volitive predicates such as wish, want, 
like, dislike, hate etc., emotive predicates such as amuse, upset, distress, 
intimidate, etc., mental perception predicates, such as know, expect, 
think, believe etc., P~ may also correspond to some other classes of pred- 
icates which we shall omit for lack of space. The common feature 
which brings together these predicates is their status of two-place 
operators: applied to an embedded sentence and then to a noun-phrase 
they yield a sentence. For example: want (come John) I, believe (come 
Mary) Nick. 
The three-place predicate Ps corresponds to causative and commu- 
nicative verbs such as order, persuade, say, tell, beg, implore. For example: 
order man (open fire man) qff~cer, tell John (come Mary) Nick. 
The above illustration of the two- and three-place predicates is 
given in the so-called hybrid language. The hybrid language is also a 
formal operator language. It answers the description given on pp. 63-66 
except for the composition of Vs. If in the operator metalanguage 
proper V s is the lexicon of symbols whose deaotata are elementary 
meanings called elementary semions (predicates, terms), in the hybrid 
language the role of semions is played by the lexical stems of a con- 
crete natural language. As a result a mixed language is obtained. Its 
grammatical component is the applicative grammar, and its lexical 
component is constituted by the lexical items of a concrete language. 3 
The embedded sentence ,~ in (14) is rewritten as either a one-place 
For greater detail see S. K. ~AUMJAI~I (1971), pp. 137-139. 
8 See S. K. ~AUMJ^N (1971), pp. 134-135. 
FORMAL METALANGUAGE AND FORMAL THEORY 71 
or a two-place predicate of the type/51 T, 152 7'1 T 2, where/5~ and/52 
are predicates with terms as their arguments (rather than embedded 
sentences as in case of/)1, P2 and P3 in (14)) or as predicates, such as 
in (14): 
/5 r 
/32TI T~ 
(15) S --->. PI~ P2ST 
P~ T~ g T~ cjg  
If the names of the predicates: modal Md, aspective As, volitive 
VI, mental perception Mr, causative Ca and communicative Cm are 
introduced in the genotype language as constants which are substi- 
tuted for the variables/)1, P2 and P3 as in (16)-(18): 
(16) P'~ As 
(17) Ps -~ Em 
Mt 
(,8, lCalcm 
a great variety of sentences can be obtained which constitute an essen- 
tial part of the basic genotype language. 
For example, a basic genotype language sentence 
(19) Md(Mt(/51TI)T ") 
is obtained by rule (14), a double application of rule (15), rules (16) 
and (17). It may be interpreted as Perhaps, you think she is wrong. A 
basic genotype language sentence 
(20) CaT~(CaT~(/5~ T~T')T~)T4 
is obtained by rule (14), a double application of rule (15) and rule (18). 
It may be interpreted as Jane 4 persuaded Mary x (that) Mary 1 order the 
pupils ~ (that) pupils~ work in the garden 3. 
72 SEBASTIAN K. "SAUMJAN-P. A. SOBOLEVA 
Note 1. The identical indices of the terms point to the identity 
of the object of causation and the subject of the situation caused. 
Note 2. R.ules (14) - (18) are but a possible example of the gener- 
ative rules of the basic genotype language. 
By applying the rules of derivation to the basic genotype language 
we obtain the derived genotype language which includes basic and 
derived genotype sentences. 
In the process of derivation an essential role is played by the com- 
binators or operators on functions. 4 
The combinators may be also called transformational operators. 
In the derivations considered below we shall use the following 
combinators: 
B - the compositor of functions. It composes the predicates of 
the matrix and the embedded sentences (Cp. "predicate -raising " 
in the transformational grammar). 
The combinator B as well as the other combinators may be applied 
both to basic and derived genotype sentences according to the rule 
(21) X(YZ) --* BXYZ 
where X is the predicate of the matrix sentence, Y is the predicate of 
the embedded sentence, and BXY is a complex predicate with Z as 
its argument. 
For example 
(22) Md(.PT)--+ BMd_~T 
illustrated in the hybrid language as 
Perhaps (come John) -+ B may come John 
and interpreted as 
Perhaps, John will come -+ John may •come 
The Compositor B is als0 used for Simulating 5 Such transformations as 
I expect John •will come -+ ! expect John to come 
On the combinators see H. B. CURRY (1958), p. 153; H. B. CURRY (1963), p. 118. 
See also S. K. ~AUMJAN (1971). 
b The role of combinators in simulating transformations and obtaining semantic 
fields is discussed in S. K. ~AUMJAN and P. A. SOBOLEVA (1973). 
FORMAL. METALANGUAGE AND FORMAL THEORY 73 
L -, the confluentor of functions with identicalarguments: 
(23) XZ (YZ) ~ LXYZ 
where X is the predicate of the matrix, Y - of the embedded sentence, 
and Z is the identical argument of both sentences, for example: 
(24) CaTI(~I T' )T ~ -+ LCa151T1T 2 
which simulates the transformation of causation: 
* Mary persuaded John, John came 
Mary persuaded John to come 
The combinator L fulfils the function, analogical to "predicate- 
raising ", "subject-raising ", and "equi-NP-deletion ", applied si- 
multaneously. 
W- the duplicator, which, applied to a function with two iden- 
tical arguments,, deletes one of them, turning a two-place function 
into a one-place function: 
(25) . XYY --+ WXY 
Rule (25) is used in the process of deriving genotype analogues 
of subjective infinitives, for example 
(26) • VI(15T1)T 1 -~ BVliST~T ~ ~ W(BVL iS)T ~ 
illustrated in the hybrid language as 
like (sing John 1) John 1 ~ B like sing John 1 John 1 
W(B like sing) John ~ 
and interprete d as 
* John likes, John sings ~ * john likes John to sing ~ 
John likes to sing 
Cv - the converter of arguments: 
(27) XZY -~ CvXYZ 
74 SEBASTIAN K. SAUMJAN- P. A. SOBOLEVA 
where X is the predicate, Y and Z are its arguments. The derivation 
with the help of Cv is analogous to the transformations" flip" (" psych- 
movement ") and passivization (see foot-note 5). 
K~ - the operator of introducing a dummy argument: 
(28) 
K~ VZXY 1 
VXY ~ Ks VXZY 
K, VXYZ 
where V is the predicate, X and Y are its arguments and Z is a dummy 
argument introduced either immediately after V (if/£1 is applied), or 
after X or Y (if Ks and/£3 are applied correspondingly), for example 
(29) VI(/3T')T2 -+ K1 VL To(#TI)T s 
which may be interpreted as 
* I like, John sings -+ I like it (that)John sings 
The operator K may be used to model the derivation of various it-con- 
structions in English. The combination of K~ and Cv is similar in effect 
to extraposition. 
U- the operator which exchanges the roles of the function and 
its argument: 
(30) XY ~ UYX 
The application of this operator will be illustrated later. The combi- 
nator U is used in the process of the derivation of the genotype ana- 
logues of verbal nouns, gerundial phrases and cleft sentences. 
H - the operator which, applied to a predicate of P~ (i = 1, 2, 3) 
type changes it into a predicate of/3 type by nominalizing an embed- 
ded sentence, for example 
(31) PST ~ H/3(Rr S)T 
t which may be interpreted e.g. as 
I believe John will come ~ I believe that John will come 
The operator R~ is a nominalizer belonging to the category A~ 
and interpreted as that or a complex of morphemes prep...'s...-ing, for 
example 
FORMAL METALANGUAGE AND FORMAL THEORY 75 
(32) Em (iST~T2)T 8 ---> HEm(R r (DT~T~))T s 
which may be given an interpretation of 
* John is amused, Harry plays the piano 
John is amused at Harry's playing the piano 
Several more combinators are used in applicative generative gram- 
mar but we shall not dwell upon them here. For lack of space we shall 
also omit their categorial status.: 
Besides the combinatory derivation rules there are rules of a dif- 
ferent kind, such as relativization, pronominalization and some others. 
Here we shall give the rule of relativization: 
(33) cj z(R (xr)r) 
Where X is the predicate of the first conjoint sentence, Z - the pred- 
icate of the second conjoint sentence and Y is the identical argument 
of both X and Z. iRA is the operator of relativization, a semion belong- 
ing to the category Al3A0~0~. The full notation of the relativizer is At3A,0~ 
Ra. The notation shows that R~ is the operator which applied• to a 
sentence, maps it into a term identifier. The interpretation of (33) is 
ungrammatical for English and further derivation rules are necessary 
to obtain the analogues of complex sentences with relative clauses or 
with participial phrases. 
Now we have sufficient means at our disposal to obtain the genotype 
analogues of (12) and (13) as well as of other complex constructions 
with all kind of complements. 
The analogue of (12) is obtained from the basic genotype sentence 
Cj (~1 TI) (~ T1 T~) by applying the rule of relativization and two com- 
binatory rules with the compositor B and duplicator W. Below the 
steps of the derivation are given in a column: 
(34) 2.-b2(R~(-b'T1)T')T2 
3. \[)~(BRAP'T'T')T ~ 
4. J52(W(BRA~)T~)T~ 
e A different way of deriving gerundial structures with the help of the combinator 
U is discussed in S. K. ~AUMJ,tW and P. A. SO~OL~.VA (1973). 
76 SEBASTIAN K. SAUMJAN- P. A. SOBOLEVA 
For the sake of clarity, derivation (34) may be repeated in the terms 
of the so-called hybrid language, i.e. a language the grammatical com- 
ponent of which is formed by applicative grammar and the lexical 
component - by the lexical items of a concrete natural language, such 
as English, for example: 
(35) 
1. Cj (burn house) (see house I) 
2. see(Ra(burn house) house) I 
3. see(BRa burn house house) I 
4. see(W(BR a burn) house) I 
The basic structure in (34) and (35) is interpreted as a semigrammatical 
sentence 
* i see a house and the house is burning. 
The second line cannot be given a grammatical interpretation, it roughly 
corresponds to a hypothetical sentence 
* I see the house is burning, a house 
The third line is obtained as a result of the application of the com- 
positor B to the operator RA and after that of BR a to p1, yielding a 
complex function BRAP ~. This complex function has two identical 
arguments T 1 and T 1. If the first argument is pronominalized (we 
omit the rule here) then the third line can be given a grammatical in- 
terpretation: 
I see a house which is burning 
The fourth step of the derivation consists in the application of the du- 
plicator W which eliminates one of the identical terms, thus yielding 
a genotype analogue of (12), i.e. 
I see a burning house. 
The derived structure (13) is_ obtained, also in four steps, the basic 
sentence being Mt(P~ T 1) T ~. See the derivation below: 
1. Mt(I51T1)T ~ " " 
(36) 2. HMt(Rr(15" T*))T ~ 
3. HMt(BP.JS~T~)T ' 
4. HMt(UTIBRflS)T ~ .... 
FORMAL METALANGUAGE AND FORMAL THEORY 77 
In terms of the hybrid language (36) is presented as follows: 
1. expect (arrive John) I 
(37) 2. H expect (R~.(arrive John)) I 
3. H expect (BR T arrive John) I 
4. H expect (U John BR r arrive) I 
Rr is the nominalizing operator with the episemion A~ i.e. R r is 
the operator which, being applied to a sentence, maps it into a term. 
The categorial status of H here is AA~A~\[3A0~A0c~ i.e. H is the operator 
which changes a two-place predicate with an embedded sentence and 
a term into a two-place predicate with two terms as its arguments. 
The first line of derivation (37) is interpreted as 
I expect John is arriving 
The second line may correspond to a sentence of the type 
I expect that John is arriving. 
The third line cannot be given a grammatical interpretation. Here the 
combinator B composes the functions Rr and P producing a composite 
function BRrP roughly corresponding to a verbal noun of the arrival 
type. To make the interpretation grammatical the combinator U is 
applied at the fourth step, turning T into a function and BRrP into 
its argument. The fourth line may be interpreted then as (13) i.e. the 
sentence I expect John's arrival. 
If we now compare the two possible ways of obtaining the genotype 
analogues of (12) and (13) we shall see that applicative grammar pro- 
vides us with the means of simulating natural languages both at the 
surface level - taxonomic description - and at the deep level. The 
application of derivation rules to the sentences of the basic genotype 
lfinguage simulates the process of transforming the deep semantic rep- 
resentation of sentences into their surface images. 
we shall now attempt to show how the application of different 
sets of derivation rules to one and the same basic genotype sentence 
generates a bundle of the derived genotype sentences which may be 
interpreted as a semantic field of phrases. Let us take several sentences 
with the emotive verb please and obtain their genotype analogues. 
The sentences are: 
(38) Mary is pleased that she found the dictionary 
(39) ? That She found the dictionary pleased Mary 
78 ,¢ SEBASTIAN K. SAUMJAN- P. A. SOBOLEVA 
{4o) (41) 
(42) 
It pleased Mary that she found the dictionary 
Mary is pleased to find the dictionary 
It pleased Mary to find the dictionary 
The genotype analogues of (38) - (42) are obtained from the basic 
genotype sentence 
(43) Em (_PTx T=)T = 
as a result of applying the derivation rules with the combinators discussed 
above, their choice and order differing in each particular case. 
The genotype analogue of (38) 7 is obtained as a result of the appli- 
cation of derivation rule (31). See (44) (below the derivations are 
given in columns, the first line being a basic sentence i.e. the sentence 
of the zero degree of derivation). 
(44) 1. Em(.PT* T 2) T 2 
2. HEm(Rr(_PTIT=)T 2) 
The genotype analogue of (39) is obtained as a result of the appli- 
cations of rules (31) and (27). See (45) 
(45) 
1. Em(_PT~T2)T 2 
2. HEm(Rr(iST*T=))T ~ 
3. Cv,2(HEm)T~(Rr(!3T'T=)) 
In derivation (45) the combinator of conversion Cv is applied 
to the result of derivation (44) which changes the places of the term 
and the embedded sentence, yielding the derived structure with a nom- 
inalized embedded sentence as its final argument (interpreted as the 
subject) (Cp. "Flip " or" psych-movement" in transformational gram- 
1Tlar). 
The genotype analogue of (40) is obtained as a result of four rules 
-(31), (28), (27). See (46) 
1. Em(~T1T~)T ~ 
(46) 2. HEm(Rr(~T'T~))T ~ 3. K(HEm)T.(R ( TIT2))T, 
4. Cvla(K(HEm))T"(Rr(-~TI T2)T=)To 
7 We do not touch upon the problem of pronominalization. This problem is discussed 
in the papers S. K. ~AUMJAN (1971) and S. K. St, uM.l,~a,q P. A. SOBOt~VA (1973). 
FORMAL METALANGUAGE AND FORMAL THEORY 79 
We shall illustrate derivation (46) in the hybrid language. See (47): 
(47) 
1. please (find dictionary Mary) Mary 
2. H please (that (find dictionary Mary)) Mary 
3. K(H please) it (that (find dictionary Mary)) Mary 
4. CvI3(K(H please))Mary (that (find dictionary Mary)) it 
The first line i.e. the basic sentence, is interpreted as 
* Mary is pleased Mary (she) found the dictionary 
The second line is interpreted as (38). 
The third line, obtained after the introduction of the dummy ar- 
gument it is interpreted as an ungrammatical sentence 
* Mary is pleased (about) it that she found the dictionary' 
The fourth line, in which the combinator Cv13 changes the places 
of the first and the third arguments gives the analogues of (40) (Cp 
" extraposition " and "flip "). 
The genotype analogue of (41) is obtained as a result of two rules - 
(21) and (25). See (48) 
(48) 
1. Em(PTxT~)T 2 
2. BEm(PT1)T2T ~ 
3. W ((BErn) (~V))T 2 
Derivation (48) is illustrated in the hybrid language. See (49): 
(49) 1. please (find dictionary Mary)Mary 2. B please (find dictionary) Mary Mary 
3. W((B please) (find dictionary) Mary 
In the second line the combinator B composes the predicates Era- 
please and ~Tl-find dictionary. The complex predicate BErn (~T ~) is 
interpreted as an infinitive complex is pleased to find the dictionary (Cp. 
"predicate raising "). In the third line the combinator W deletes the 
identical terms T~ Mary, yielding (41). 
The analogue of (42) is obtained in five steps: to derivation (48) 
two more lines are added (rules (28) and (27)). See (50): 
80 SEBASTIAN K. SAUMJAN- P. A. SOBOLEVA 
(50) 4. K(W((BEm) (.~T')))ToT2 5. Cv,3(K(W((B~m) (~rl))))r, To 
The set of derivations from the basic sentence (43) may be shown 
in the tree below. See (51): 
(51) 
EmST 
• H B 
Cvi~ (38) 
(39)~ 
(*)~ 
CIJl8 
(4o) 
(*) 
W 
(41) 
K 
(*) 
CI,123 
(42) 
At the root of the tree we have a basic genotype sentence. The 
lines indicate the combinators which are applied to the basic or deriv- 
ed sentences in our derivations. The numbers (38) - (42) refer to the 
sentences which interpret the derived genotype sentences, (*) denotes 
the absence of" a grammatical interpretation in English. Similar trees 
of derivation may be drawn for other basic sentences of the genotype 
language. 
Examples of such trees for the basic sentences MdS, Asg, MtST, 
VI~T, CaTI~T ~ and CmTI~T 2 may be found in the paper of S. K. 
~AUMJAN and P. A. SOBOrEVA (1973). 
The basic genotype sentence at the root of the tree may be understood 
as a generalized semantic representation, which we call a standard mac- 
rosituation. The derived genotype sentences at the other nodes are 
semantic representations called derived macrosituati0ns. Interpreta- 
tions (38) - (42) are concrete semantic representations called micro- 
situations. 
The" tree in (51) may be regarded as a possible scheme of derivation 
FORMAL METAI, ANGUAGE AND FORMAL THEORY 81 
of the emotive semantic field in English mapped into the genotype 
language. 
The purpose of (34) - (49) was to show that the combinators may 
be used to form a unified system of derivations necessary for the construc- 
tion of the derived genotype language. In (38) - (49) the genotype 
sentences werederived from one and the same basic genotype sentence, 
differing from each other only in the choice, number and order Of ap- 
plication of the combinators. Hence, the possibility of simulating the 
hierarchical organisation of grammatical synonymy as well as the proc- 
ess of embodying thought in linguistic forms. 
Let us now turn to the definition of formal theories of natural 
languages. 
The formal theory of natural languages is a formal system which 
distinguishes between the following components: 
1. The basic genotype language is defined by the schemes of 
sentences serving as semantic representations. Each semantic represen- 
tation corresponds to a definite situation. Semantic representations are 
formed from elementary semions. Elementary semions are interpreted 
as elementary meaning units. Only two types of elementary semions 
are allowed - elementary predicates and elementary terms - and.two 
types of sentences: 1) sentences with one-, two- and three-place ele- 
mentary predicates and an embedded sentence as one of their argu- 
ments, 2) sentences which consist of sentences connected by means 
of a conjunctor. 
2. The rules of derivation of the derived genotype language sen- 
tences from the sentences of the basic genotype language. The sentences 
of the basic genotype language are regarded as the sentences of the 
zero degree of derivation, hence the basic genotype language is regard- 
ed as part of the derived genotype language which is a sum of the 
zero degree of derivation sentences. E~)ery rule of derivation of the 
derived genotype language sentences is a substitution of some expres- 
sion containing combinators applied to the elementarypredicates instead 
of some expression of the basic genotype language. The rules of deri- 
vation of the derived genotype language may be called combinatory 
rules because they consist in the substitution of the expressions with 
the various types of combinators applied to elementary predicates in 
place of the basic genotype language expressions. 
3. The rules of derivation of phenotype language sentences from 
sentences of the derived genotype language. These rules may be called 
phenotype rules. Every phenotype rule is a substitution of: a) a certain 
82 V SEBASTIAN K. SAUMJAN-P. A. SOBOLEVA 
expression comprising phenotype semions into a certain expression of 
the derived genotype language. Or: b) into a certain phrase obtained 
according to (a). The phenotype rules meet the requirement according 
to which any application of the phenotype rule must not break the 
applicative structure of the sentence. Sentences obtained as a result of 
the application of the phenotype rules are called phenotype sentences. 
4. The rules of linear transformations. 
In the genotype language expressions have a standard linear structure: 
an operator precedes its operand. This standard structure is retained by 
the phenotype sentences. The aim of linear transformations is to con- 
vert the standard structure into a linear structure characteristic of the 
sentences of the corresponding natural language. 
The rules of linear transformations are based on the use of two com- 
binators of permutations. The use of the permutation combinators is 
interesting from the typological point of view, since the standard linear 
structure may be regarded as a typological standard, the derivation 
from which may be accounted for by different sets of permutation 
combinators. 
5. Morpho - phonological rules. 
Morpho - phonological rules consist in the substitution of either 
phonological strings instead of phenotype expressions, or of some pho- 
nological strings instead of other phonological strings. 
In accordance with the components of the formal theory of natural 
languages considered above the scheme of deduction in this theory 
looks as follows: 
G1 .... , G,F1, ..., FmL1, ..., Lp01 ..... Oq 
G1 is a sentence of the basic genotype language. G, is a sentence of the 
derived genotype language. In case n = 1, G~ is a sentence of the zero 
degree of derivation. F1 is the initial sentence of the phenotype compo- 
nent. F,~ is the final sentence of the phenotype component. L1 is a 
sentence obtained at the first step of the application of the linear transfor- 
mational rules. Lp is a sentence obtained at the final step of the linear 
transformation rules. 
O1 a morpho-phonological representation of the sentence obtained 
at the first step and Oq - at the last step of the application of the morpho- 
phonological rules. 
The formal theory of natural languages distinguishes between the 
concept " macrosituation " and the concept " microsituation ". A mac- 
FORMAL METALANGUAGE AND FORMAL THEORY 83 
ros~tuation is an abstract situation at the level of generalized semantic 
representation. The microsituations pertaining to the given macro- 
situations are its various interpretations at the level of concrete semantic 
representations. An example of an " emotive" macrosituation: "An 
animate passive subject (patient) experiences an emotion in connection 
with the action it performed ". This macrosituation may be realized 
in the following microsituations: Mary is pleased that she found the dic- 
tionary, John is upset that he lost the pen, Jane is confused that she upset the 
cup etc. An example of a causative macrosituation: " An animate active 
subject (agent) causes an animate object (patient) to perform an action " 
This macrosituation may be realized in the following micrositua- 
tions: The officer ordered the men to open fire, The mother persuaded her 
son to take a walk, The teacher makes the pupils work etc. 
In connection with the distinction between macro and micrositua- 
tions we distinguish between two levels of derivation in the formal 
theory of natural languages: derivations at the level of macrosituations 
and derivations at the level of microsituations. As to the formal theory 
of natural languages it must provide for the rules of derivation at the 
level of macrosituations. The level of microsituation is an interpretative 
level which is a function of a special component which may be called 
the dictionary of concrete lexemes. The dictionary of concrete lexemes 
consists of dictionary entries corresponding to separate concrete lexemes. 
Every dictionary entry must contain the necessary grammatical 
information providing for its inclusion as a concrete semion interpret- 
ant of the macrosituation in a derivation set. 
In conclusion we shall consider the concept "semantic fields ". In 
connection with the semantic field we shall split the concept "macro- 
situation" into the concept "standard macrosituation " and "derived 
macrosituation ". The standard macrosituation is a semantic repre- 
sentation corresp ondin g to the sentences of the basic g eno typ elan g ua g.e 
For example Em(iST 1 T ~) T ~ which roughly corresponds to the "emo- 
tive situation" cited above or CaT 1 (P T~ T ~) T 8 which is very close 
to the above causative situation. However, the latter can be given a 
more exact interpretation: "an animate subject (ager~t) brings pressure 
to bear on an animate object (patient) and the latter performs an 
action" 
The derived macrosituation is a semantic representation correspond- 
ing to the sentences of the derived genotype language. For example 
HEm (Rr(i 5 T 1 T~) ) T 2 which exactly corresponds to the above "emo- 
tive" situation of L Ca(iST2)T ~ T s which exactly corresponds to the 
84 SEBASTIAN K. ~AUMJAN- P. A. SOBOLEVA 
"causative" situation in its previous formulation, i.e. "an animate 
active subject causes an animate passive object to perform an action" 
In the formal theory of natural languages considered above the se- 
mantic field is a bundle of derivations simulating a set of derived mac- 
rosituations from a basic genotype language sentence which simu- 
lates a standard macrosituation. See (51). The standard and the derived 
macrosituations are in relations of meaning inclusion. 
In accordance with the destination between macro- and microsit- 
uations we distinguish between the semantic fields at the level of 
macrosituations and the semantic fields at the level of microsituations: 
macrosemantic fields and microsemantic fields. Each macrosemantic 
field corresponds to a finite set of microsemantic fields which are its 
interpretations. 
On the basis of what has been considered above it is possible to 
draw the following general conclusions pertaining to any generative 
grammar. 
The term" generative grammar" must be understood in two senses: 
as the grammar of a formal metalanguage used by the formal theory 
of natural languages and as the formal theory of natural languages 
proper. 
A formal theory of natural languages must be based on the two- 
level principle which distinguishes between the two levels of abstraction: 
the phenotype and genotype levels. The genotype level is not given 
in immediate observation but is postulated as a hypothetical object 
which is a theoretical construct. 
It is necessary to distinguish between the two levels of formal der- 
ivation: the level of macrosituations and the level of microsituations. 
Grammar as a formal theory of natural languages has to do with mac- 
rosituations. As to the level of microsituations it has to do with a 
special dictionary which is compiled for the needs of the generative 
grammar. 

REFERENCES 

H. B. CURRY, Combinatory logic, vol. 1, 
Amsterdam, 1958. 

H. B. CURRY, Foundations of mathematical 
logic, New York, 1963. 

S. K. ~AUMJAN, Genotipi~eskij jasyk i 
formalnaja semantika, in , Problemy 
structurnoj linguistiki ~, 1971. 

S. K. ~AUMJAN, P. A. SOBOtEVa, Kom- 
plementacija v aptflicativno j grammatike, 
in ~ Problemy structurnoj linguistiki ~, 
1973. 
