2. Mourdoukhay-Boltovskoy D. Sur les Syllogismes
en logique et les Hypersyllogismes en Metalogique
// Proceedings of Naturalist Society of NKSU.
Vol.3. Rostov-on-Don, 1919-1926. P. 34-35.
Metalogics is constructed which relates to classical
logics the same manner four-dimensional space
relates to the usual space. Laws of formal logic of
propositions are preserved and laws of logic of
classes are replaced with the more general ones. A
hyperproposition is the relation of not two but three
terms
| a” b’ c |
of a species, a genus and a hypergenus.
Hyperclass presupposes not one dual (contrary)
hyperclass but two (a), ( a ), and not two operations
but three:
( a ) = a, ( ā ) = ( a ).
3. Mourdoukhay-Boltovskoy D. Sur les Syllogismes en logique et les
Hypersyllogismes en Metalogique // Proceedings of Naturalist Society of NKSU.
Vol.3. Rostov-on-Don, 1919-1926. P. 34-35.
It is necessary to introduce a general negative
hyperproposition:
|a b c|
| a’’ b’ c | = | a ( ̅b) (c) |
| a b c | = | a ( ̅b ) (c) | (obversio)
A partial affirmative hyperproposition contains 6 terms:
ea fb gc |
and reduces to the claim of the existence of such a
hyperclass х that:
x' ' e' a
x' ' f' b
x' ' g' c
4. Mourdoukhay-Boltovskoy D. Sur les Syllogismes en logique et les
Hypersyllogismes en Metalogique // Proceedings of Naturalist Society of NKSU.
Vol.3. Rostov-on-Don, 1919-1926. P. 34-35.
And in virtue of the preservation of the laws of
propositional logic
| ea fb gc | | fb ea gc | ... (conversio)
Negative propositions are
x' ' e' a
| ea fb gc | x' ' f' b
x' ' g' с
x' ' e' a
| ea fb gc | x' ' f' b
x' ' g' с
5. “Before me the notion of
Metalogics was elaborated just
from a philosophical and not a
mathematical point of view by
prof.N. Vasiliev”
[Mourdoukhay-Boltovskoy D.D.
Philosophy. Psychology.
Mathematics. Moscow, 1998.
p.488]
6. Taking this into consideration then introduced
by Mourdoukhay-Boltovskoy notion of
hyperproposition as the relation of not two but
three terms | a” b’ c | – a species, a genus
and a hypergenus would be tentatively treated
as “any a is b in all (imaginary) worlds and
especially is c in some distinguished
(imaginary) worlds”.
In this case it becomes clear why “hyperclass
presupposes not one dual (contrary)
hyperclass but two (a), ( a ), and not two
operations – inclusion and exclusion – but
three: ( a ) = a, ( ā ) = (a)”
Here (a) rather should be treated as a
complementation to a in some specific worlds.
7. ( a ) = a follows from that
considering first (contrary)
a complementation to a hyperclass
in one world we then considering a
complementation to this complementation
in all worlds thereby returning
to initial a (taking into account
contrarity of the hypergenus
complementation).
( ā ) = (a) then follows from that taking
initially a complementation to a in one
world we then take a complementation to a
in some other world but since that world is
chosen arbitrary then it intends
complementation in all worlds.
This would be illustrated with the help of
topological operations of interior and
boundary for classes.
8. HYPERDIAGRAMS
• A hyperclass a would be
The dual hyperclass a
sketched out with a help (a Boolean complementation)
of the following diagram:
The dual hyperclass a
(a hypercomplementation)
14. The fundamental translation of
hypersyllogistic into predicate calculus
| a” b’ c | x((A(x) B(x)) (A(x) C(x)) (C(x) B(x)))
|a b c| x((A(x) B(x)) (A(x) C(x)) ( C(x)
B(x)) )
15. How hypersyllogistic would be
semantically linked with Vasiliev’s
imaginary logic
T.P.Kostyuk “N.A.Vasiliev’s N-dimensional Logic: Modern Reconstruction”
<D, , 1, 2, 3>
where D ,
(v) D,
1, 2, 3 –functions assigning to any general term P subsets of D
having the following properties:
1(P) , 1(P) 2(P) = ,
1(P) 3(P) = ,
2(P) 3(P) = ,
1(P) 2(P) 3(P) = D.
From informal point of view 1(P) is treated as a volume, 2(P) as anti-
volume and 3(P) as contradictory domain of the term P.
16. How hypersyllogistic would be semantically linked
with Vasiliev’s imaginary logic
| a” b’ c | = 1 1(a) 1(b) & 1(a) 1(c) & 1(c) 1(b)
|a b c|=1 1(a) 2(b) & 1(a) 3(c) & 3(c) 2(b)
