(Tbilisi
State University)
From
the simplified formal point of view the natural language is a set of words. We
consider the word as a well formed expression of natural language. Let us assume
that W is a language i.e. W is a set of words. Logical-semantical analysis of
language show that from W we can separate the w subset of
the basic words of this language. We call the elements of W-w set as
non-basic or definable or contracted words of the language, e.g. “I lie” is
a non-basic word, because “I lie” means “I say something (A) and I know
that this something (A) is not true”. The formal-logical description of the
natural language shows us that “I lie” must be regarded as 1-placed operator
which operates on l type words and gives l type words. This means that “I
lie” is l→l type word and its meaning is the truth scheme. So, to define
it as true or false is absurd and gives the paradox.
Very briefly about
that groundings by which we think that it is possible to organize the natural
language mathematically according to the strong formal-logical requirements.
Let us assume To
is a formal theory and L(To) is its language which consists of s1n1,…,
sknk
basic words and –1, –2 ,… symbols. Among the s1n1,…,
sknk
at least one is 0-place basic word.
Definition: The expression (ska1…ak)n
is called n-place word (n³0)
if ai
is a word or ai
is – j (jÎ1, –n) and if any ¢–
j¢ (jÎ1,
–n) has at least one occurrence in (ska1…ak)n.
Definition: I(To) is an interpretation of To
theory if it is a consistent (non-contradictional) /\-set of (aka1…ak)o
= a
type expresssions, where (aka1…ak)o
is a
0-placed word of L(To) language and a is either t or f. I(To) is complete if any 0-placed
word of L(To) language is interpreted within I(To).
In any complete I(To)
interpretation any 0-placed word has its own truth value. In the non-complete
I(To) interpretations the truth value of some 0-placed words is i
(indefinite). This means that we are in such interpretation world that we do not
know strictly the truth value of this word, although we know that 0-placed word
in this interpretation is either t or f.
Now, let us assume
that T1 theory is derived from To theory by its extension
through new non-basic i.e. contracted P1n1
word according to the following d1 contracting rule
d1:
P1n1 – 1
… – n1 ––– (W(To)) n1
(Here P1n1
is knew n1-place word-operator and (W(To)) n1
is any n-place word of To theory. The meaning of the symbol “–––”
is “formally means”. This symbol together with the d1 rule can be
regarded as the bridge between the To and T1 theories.)
Analogously d2 contracting rule gives T2 theory from T1
theory and so on we can receive new extended Tk+1 theory from the
previous Tk theory by the dk+1 contracting rule.
De|(W(Tk))n| is the designation of the
reduction of (W(Tk))n word of Tk theory into Te
theory. Such reduction is a result of step by step exlusion of new non-basic
contracted Pknk
,…, Pe+1ne+1
words, which is made according to the dk…dk+1 rules.
It is clear that De|W(Tk)n|
is n-placed word of Te theory.
R a1…an
(W(Tk))n (Rea1…an
(W(Tk))n is derived by the replacement of ai
(De|ai|)
word within (W(Tk))n (De|(W(Tk))n|)
instead of – i symbol.
It is proved, that
Rea1…an(W(Tk))n
=.
De|Ra1…an(W(Tk))n|
= RDe|a1|…De|an|De|(W(Tk))n|.
By {£. (W(Tk))n}em we marked the set of all m-place specific cases of (W(Tk))n word in Te
theory. It is clear, that elements of {£.
(W(Tk))n}00
are R0a10…an0(W(Tk))n
type expressions.
Above we have
mentioned in a simplified form some methods and ideas of notation theory.
In the article are
proved that if T0 theory is non-contradictional then
non-contradictional is any Tk theory which is formed according above
mentioned rules.
Moreover: if within
Tk theory we make an assumption that (W(Tk))0 =
a(aÎ{t,f}) and as a result we received contradictional
statement, we can conclude (in a storing formal sense) that (W(Tk))0
= –a is
proved.
Consequently, we
conclude that by suggested formal way it is possible to build Hilbertian
non-contradictional and non-paradoxical formalizm.
We distinguish A«ùA
and A–ùA expressions. The former is trivial contradictional
logical schemes. The later is considered as the trivial paradoxical scheme. We
think that any paradox can be reduced directly or indirectly to the trivial
paradoxical scheme*. Even if it rised on the base of contradictional
pseudo-basic theory**. By our point of view the theory is (really) basic (i.e.
non-pseudo-basic) if its interpretable
symbols get their value in any interpretation separately of each other.
Therefore, (really) basic theory in any interpretation gives non-contradictional
theory. So, if the theory is contradictional it cannot be basic!
The
question how and when it is possible to reduce the pseudo-basic theory to the
basic theory is an important mathematical question. We suppose that for the
natural language as a formal theory we can find its basic theory.
Let
us assume that d: A – B rule defines some C new word in A expression (B is a
word (form) of the T theory). This C word put T theory into new Tc
theory if {£.
A}oTc
and {£. B} oTc
sets consist of contral expressions (P and ùP
is general face of contral expression) i.e. if {£. A} oTc
Çù
{£.
B} oTc ¹
Æ
then Tc theory is paradoxical and the reason of this paradoxizm is
the abovemantioned d: A – B rule.
Such situtaion observed in Russel’s paradox. Russel’s R set is defined as
new word by rule – 1 Î
R ––
– 1 Ï
– 1. This rule take us from T theory into TR theory,
where RÎRÎ{£.
(– 1 Î
R)} and RÏRÎ{£. (– 1Ï–
1)}. So, under any interpretation of the meanings of R and Î symbols if Ï simbol is defined as – 1Ï–
2 ––– ù(–
1Ζ
2) then we have TR paradoxical theory. Analogous reason exists
in Greling’s semantic paradoxes but there are violated also other rule of the
logic of language similarly to lie-paradox.
**
Assume, To is pseudo-basic theory, L(To) = {s11 ,
s21 , ao,
bo} and (s11 ao)«(s21
bo) is restriction axiom of this pseudotheory. Then due to sk1ao
––– sk–1bo
rule ({k,–k} = {1,2}) To pseudo-basic
theory reduced on Tk basic theory (in this case in To
theory sk1
is partly contracted symbol it means that sk1
in To theory is partly briefly interpreted symbol).
Thus
we think that the above mentioned formal way which are made on the bases of
Frege’s compositional principle and notation theory seems to be affective for
a formal-logical representation of natural language, where Rassell’s type
theory plays an important role.
Reference
1. Bentem
J, Essays on Logical Semantics, 1997; Logic and Language, Handbook, 1998.
2. Frege
G, Beguffsschrift, 1879.
3. Russel
B, The Principles of Mathematics, 1938; Ludwig Wittgenstein, 1951.
4.
Wievtbicka A, Semantics Primes and Universals, Oxford University, Press,
1996.
5.
Wittgenstein L, Note books, 1914-1916, 1961.
6. Ãèëüáåðò
Ä, Òåîðèÿ äîêàçàòåëüñòâ, 1972.
7. Ïõàêàäçå
Ø, Íåêîòîðûå âîïðîñû òåîðèé îáîçíà÷åíèé, 1978.
.