Vladimir L.VasyukovInstitute of PhilosophyRussian Academy of SciencesMoscow, Russia
Mourdoukhay-Boltovskoy D. Sur les Syllogismesen logique et les Hypersyllogismes en Metalogique   // Proceedings of Natural...
Mourdoukhay-Boltovskoy D. Sur les Syllogismes en logique et lesHypersyllogismes en Metalogique // Proceedings of Naturalis...
Mourdoukhay-Boltovskoy D. Sur les Syllogismes en logique et lesHypersyllogismes en Metalogique // Proceedings of Naturalis...
“Before me the notion ofMetalogics was elaborated justfrom a philosophical and not amathematical point of view byprof.N. V...
Taking this into consideration then introducedby Mourdoukhay-Boltovskoy notion ofhyperproposition as the relation of not t...
( a ) = a follows from thatconsidering first (contrary)a complementation to a hyperclassin one world we then considering a...
HYPERDIAGRAMS• A hyperclass a would be                              The dual hyperclass a  sketched out with a help    (a ...
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...
How hypersyllogistic would be     semantically linked with Vasiliev’s             imaginary logicT.P.Kostyuk “N.A.Vasiliev...
How hypersyllogistic would be semantically linked         with Vasiliev’s imaginary logic| a” b’ c | = 1   1(a)   1(b) &  ...
To be continuedThank you foryour attention
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Vasiliev mordukhai

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Vasiliev mordukhai

  1. 1. Vladimir L.VasyukovInstitute of PhilosophyRussian Academy of SciencesMoscow, Russia
  2. 2. Mourdoukhay-Boltovskoy D. Sur les Syllogismesen 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. 3. Mourdoukhay-Boltovskoy D. Sur les Syllogismes en logique et lesHypersyllogismes 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. 4. Mourdoukhay-Boltovskoy D. Sur les Syllogismes en logique et lesHypersyllogismes 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. 5. “Before me the notion ofMetalogics was elaborated justfrom a philosophical and not amathematical point of view byprof.N. Vasiliev”[Mourdoukhay-Boltovskoy D.D.Philosophy. Psychology.Mathematics. Moscow, 1998.p.488]
  6. 6. Taking this into consideration then introducedby Mourdoukhay-Boltovskoy notion ofhyperproposition as the relation of not two butthree terms | a” b’ c | – a species, a genusand a hypergenus would be tentatively treatedas “any a is b in all (imaginary) worlds andespecially is c in some distinguished(imaginary) worlds”.In this case it becomes clear why “hyperclasspresupposes not one dual (contrary)hyperclass but two (a), ( a ), and not twooperations – inclusion and exclusion – butthree: ( a ) = a, ( ā ) = (a)”Here (a) rather should be treated as acomplementation to a in some specific worlds.
  7. 7. ( a ) = a follows from thatconsidering first (contrary)a complementation to a hyperclassin one world we then considering acomplementation to this complementationin all worlds thereby returningto initial a (taking into accountcontrarity of the hypergenuscomplementation).( ā ) = (a) then follows from that takinginitially a complementation to a in oneworld we then take a complementation to ain some other world but since that world ischosen arbitrary then it intendscomplementation in all worlds.This would be illustrated with the help oftopological operations of interior andboundary for classes.
  8. 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)
  9. 9. Hyperproposition | a” b’ c |
  10. 10. General negative hyperproposition | a b ̅c || a b ̅c | = | b c ̅a | = | c a ̅b |
  11. 11. Particular affirmative hyperproposition
  12. 12. Negative hyperproposition
  13. 13. Negative hyperproposition
  14. 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. 15. How hypersyllogistic would be semantically linked with Vasiliev’s imaginary logicT.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. 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)
  17. 17. To be continuedThank you foryour attention

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