Modal Environment for Boolean Speculations

preliminary report
  • George Gargov
  • Solomon Passy
  • Tinko Tinchev


The common form of a mathematical theorem consists in that “the truth of some properties for some objects is necessary and/or sufficient condition for other properties to hold for other objects”. To formalize this, one happens to resort to Kripke modal logic K which, having in the syntax the notions of ‘property’ and ‘necessity’, appears to provide a reliable metamathematical fundament. In this paper we challenge this reliability. We propose two different approaches each claiming better formal treatment of the state of affairs. The first approach is in formalizing the notion of ‘sufficiency’ (which remains beyond the capacities of K), and consequently of ‘sufficiency’ and ‘necessity’ in a joint context. The second is our older idea to formalize the notion of ‘object’ in the same modal spirit. Having ‘property, object, sufficiency, necessity’, we establish some basic results and profess to properly formalize the everyday metamathematical reason.


Modal Logic Kripke Model Dynamic Logic Completeness Theorem Modal Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. van Benthem, J.F.A.K., 1977, Modal Logic as Second-Order Logic, Report 77-04, Dept. of Mathematics, Univ. of Amsterdam, March.Google Scholar
  2. van Benthem, J.F.A.K., 1979, Minimal Deontic Logics (abstract), Bull, of Sec, of Logic, 8, No.1 (March), 36–42.zbMATHGoogle Scholar
  3. van Benthem, J.F.A.K., 1984, Possible Worlds Semantics: A Research Prorgam that Cannot Fail?, Studia Logica, 43, No.4, 379–393.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Bull, R.A., 1968, On Possible Worlds in Propositional Calculi, Theoria, 34.Google Scholar
  5. Goldblatt, R.I., 1974, Semantic Analysis of Orthologic, J. Philos. Logic, 3, 19–35.MathSciNetCrossRefGoogle Scholar
  6. Goldblatt, R.I., 1982, Axiomatizing the Logic of Computer Programming, Springer LNCS 130, Berlin.Google Scholar
  7. Goldblatt, R.I. & S.K. Thomason, 1975, Axiomatic Classes in Propositional Modal Logic, in: Springer LNM 450, 163–173.Google Scholar
  8. Humberstone, I.L., 1983, Inaccessible Worlds, Notre Dame J. of Formal Logic, 24, No.3 (July), 346–352.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Humberstone, I.L., 1985, The Formalities of Collective Omniscience, Philos. Studies 48, 401–423.MathSciNetGoogle Scholar
  10. Passy, S.I., 1984, Combinatory Dynamic Logic, Ph.D. Thesis, Mathematics Faculty, Sofia Univ., October.Google Scholar
  11. Passy, S. & T. Tinchev, 1985a, PDL with Data Constants, Inf. Proc. Lett., 20, No.1, 35–41.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Passy, S. & T. Tinchev, 1985b, Quantifiers in Combinatory PDL: Completeness, Definability, Incompleteness, in: Springer LNCS 199, 512–519.Google Scholar
  13. Rasiowa, H. & R. Sikorski, 1963, Mathematics of Metamathematics, PWN, Warsaw.zbMATHGoogle Scholar
  14. Segerberg, K., 1971, An Essay in Classical Modal Logic, Uppsala Univ.zbMATHGoogle Scholar
  15. Tehlikeli, S. (S. Passy), 1985, An Alternative Modal Logic, Internal Semantics and External Syntax (A Philosophical Abstract of a Mathematical Essay), manuscript, December.Google Scholar
  16. Tinchev, T.V., 1986, Extensions of Propositional Dynamic Logic (in Bulgarian), Ph.D. Thesis, Mathematics Faculty, Sofia Univ., June.Google Scholar
  17. Vakarelov, D., 1974, Consistency, Completeness and Negation, in: Essays on Paraconsistent Logics, Philosophia Verlag, to appear.Google Scholar
  18. Weyl, H., 1940, The Ghost of the Modality, in: Philosophical Essays in Memory of Edmund Husserl, Cambridge (Mass.), 278–303.Google Scholar

Copyright information

© Plenum Press, New York 1987

Authors and Affiliations

  • George Gargov
    • 1
  • Solomon Passy
    • 1
  • Tinko Tinchev
    • 1
  1. 1.Faculty of Mathematics at Sofia UniversitySofiaBulgaria

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