Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
van Benthem, J.F.A.K., 1977, Modal Logic as Second-Order Logic, Report 77-04, Dept. of Mathematics, Univ. of Amsterdam, March.
van Benthem, J.F.A.K., 1979, Minimal Deontic Logics (abstract), Bull, of Sec, of Logic, 8, No.1 (March), 36–42.
van Benthem, J.F.A.K., 1984, Possible Worlds Semantics: A Research Prorgam that Cannot Fail?, Studia Logica, 43, No.4, 379–393.
Bull, R.A., 1968, On Possible Worlds in Propositional Calculi, Theoria, 34.
Goldblatt, R.I., 1974, Semantic Analysis of Orthologic, J. Philos. Logic, 3, 19–35.
Goldblatt, R.I., 1982, Axiomatizing the Logic of Computer Programming, Springer LNCS 130, Berlin.
Goldblatt, R.I. & S.K. Thomason, 1975, Axiomatic Classes in Propositional Modal Logic, in: Springer LNM 450, 163–173.
Humberstone, I.L., 1983, Inaccessible Worlds, Notre Dame J. of Formal Logic, 24, No.3 (July), 346–352.
Humberstone, I.L., 1985, The Formalities of Collective Omniscience, Philos. Studies 48, 401–423.
Passy, S.I., 1984, Combinatory Dynamic Logic, Ph.D. Thesis, Mathematics Faculty, Sofia Univ., October.
Passy, S. & T. Tinchev, 1985a, PDL with Data Constants, Inf. Proc. Lett., 20, No.1, 35–41.
Passy, S. & T. Tinchev, 1985b, Quantifiers in Combinatory PDL: Completeness, Definability, Incompleteness, in: Springer LNCS 199, 512–519.
Rasiowa, H. & R. Sikorski, 1963, Mathematics of Metamathematics, PWN, Warsaw.
Segerberg, K., 1971, An Essay in Classical Modal Logic, Uppsala Univ.
Tehlikeli, S. (S. Passy), 1985, An Alternative Modal Logic, Internal Semantics and External Syntax (A Philosophical Abstract of a Mathematical Essay), manuscript, December.
Tinchev, T.V., 1986, Extensions of Propositional Dynamic Logic (in Bulgarian), Ph.D. Thesis, Mathematics Faculty, Sofia Univ., June.
Vakarelov, D., 1974, Consistency, Completeness and Negation, in: Essays on Paraconsistent Logics, Philosophia Verlag, to appear.
Weyl, H., 1940, The Ghost of the Modality, in: Philosophical Essays in Memory of Edmund Husserl, Cambridge (Mass.), 278–303.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 Plenum Press, New York
About this chapter
Cite this chapter
Gargov, G., Passy, S., Tinchev, T. (1987). Modal Environment for Boolean Speculations. In: Skordev, D.G. (eds) Mathematical Logic and Its Applications. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0897-3_17
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0897-3_17
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4612-8234-1
Online ISBN: 978-1-4613-0897-3
eBook Packages: Springer Book Archive