Studia Logica

, Volume 47, Issue 4, pp 319–326 | Cite as

Mathematics of Totalities: an alternative to mathematics of sets

  • Herman Dishkant
Article
  • 20 Downloads

Abstract

I dare say, a set is contranatural if some pair of its elements has a nonempty intersection. So, we consider only collections of disjoint nonempty elements and call them totalities. We propose the propositional logicTT, where a proposition letters some totality. The proposition is true if it letters the greatest totality. There are five connectives inTT: ∧, ∨, ∩, ⌉, # and the last is called plexus. The truth of σ # π means that any element of the totality σ has a nonempty intersection with any element of the totality π. An imbeddingG of the classical predicate logicCPL inTT is defined. A formulaf ofCPL is a classical tautology if and only ifG(f) is always true inTT. So, mathematics may be expounded inTT, without quantifiers.

Keywords

Mathematical Logic Computational Linguistic Nonempty Intersection Proposition Letter Classical Predicate 

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References

  1. [1]
    R. Descartes,Regulae ad directionem ingenii, 1628.Google Scholar
  2. [2]
    D. H. H. Ingalls,Matherials for the Study of Navya-Nyaya Logic, Cambridge, 1951.Google Scholar
  3. [3]
    H. Rasiowa andR. Sikorski,The Mathematics of Metamathematics, Warszawa, 1963.Google Scholar

Copyright information

© Polish Academy of Sciences 1988

Authors and Affiliations

  • Herman Dishkant
    • 1
  1. 1.Theoretical Problems DepartmentThe USSR Academy of SciencesMoscow

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