Vasiliev’s C

lue to

Mourdoukhay-

Boltovskoy’s

Hypersyllogisti

c

Vladimir L.VasyukovInstitute of Philosophy Russian Academy of SciencesMoscow, Russia

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 ).

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 bc |

| a’’ b’c | = | a ( �b) (c) |

| a b c | = | a ( �b ) (c) | (obversio)

A partial affirmative hyperproposition contains 6 terms:

and reduces to the claim of the existence of such a

hyperclass х that:

|gcfbea

cgx

bfx

aex

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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

(conversio)

Negative propositions are

...||||

gceafbgcfbea

сgx

bfx

aex

gcfbea

сgx

bfx

aex

gcfbea

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'''

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“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]

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.

(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.

HYPERDIAGRAMS

• A hyperclass a would be sketched out with a help of the following diagram:

The dual hyperclass a (a Boolean complementation)

The dual hyperclass a (a hypercomplementation)

Hyperproposition | a” b’ c |

General negative hyperproposition | a b �c |

| a b �c | = | b c �a | = | c a �b |

Particular affirmative hyperproposition

Negative hyperproposition

Negative hyperproposition

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 bc | x((A(x) B(x)) (A(x) C(x)) (C(x) B(x)) )

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.

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 bc | = 1 1(a) 2(b) & 1(a) 3(c) & 3(c) 2(b)

Thank you for your attention

To be continued