Abstract
The purpose of this paper is to give a survey of the relational formalization of modal logics. The paradigm ‘formulas are relations’ leads to the development of a relational logic based on algebras of relations. The logic can be viewed as a generic logic for the representation of nonclassical logics; in particular a broad class of multimodal logics can be specified within its framework. As a consequence, proof systems for the relational logic become a convenient tool for the development of a proof theory for nonclassical logics. The relational logic enables us to represent within a uniform formalism the three basic components of any propositional logical system: syntax, semantics and deduction apparatus. The essential observation, leading to a relational formalization of logical systems, is that a standard relational structure (a Boolean algebra with a monoid) constitutes a common core of a great variety of nonclassical logics. Exhibiting this common core on all the three levels of syntax, semantics and deduction, enables us to create a general framework for representation, investigation and implementation of nonclassical logics.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
S. Demri, E. Orlowska. Logical analysis of demonic nondeterministic programs. Submitted, 1995.
S. Demri, E. Orlowska, I. Rewitzky. Towards reasoning about Hoare relations, Annals of Mathematics and Artificial Intelligence, 12, 265–289, 1994.
M. Herment, E. Orlowska. Handling information logics in a graphical proof editor. Computational Intelligence, 11, 297–322, 1995.
C. A. R. Hoare, He Jifeng. The weakest prespecification, Fundamenta Informaticae, 9, Part I: 5184, Part II: 217–262, 1986.
S. Kripke. Semantical analysis of modal logic I. Zeitschrii t fiir Mathematische Logik and Grundlagen der Mathematik, 9, 67–96, 1963.
S. Kripke. Semantical analysis of intuitionistic logic. In: J. N. Crossley, M. A. Dummett, (eds.), Formal Systems and Recursive Functions, North Holland, Amsterdam, 1965.
R. Maddox. The origin of relation algebras in the development and axiomatization of the calculus of relations. Studia Logica, 50, 421–456, 1991.
L. Maximova, D. Vakarelov. Semantics for w+-valued predicate calculi. Bulletin of the PAS, Ser. Math., 22, 756–771, 1974.
D. Monk. On representable relation algebras. Michigan Mathematical Journal, 11, 207–210, 1964.
D. Monk. Nonfinitizability of classes of representable cylindric algebras. Journal of Symbolic Logic, 34, 331–343, 1969.
E. Orlowska. Logic of nondeterministic information. Studia Logica, 44, 93–102, 1985.
E. Orlowska. Relational interpretation of modal logics. In: H. Andreka, D. Monk, I. Nemeti, (eds), Algebraic Logic. Colloquia Mathematica Societatis Janos Bolyai 54, North Holland, Amsterdam, 443–471, 1988.
E. Orlowska. Relational proof system for relevant logics. Journal of Symbolic Logic, 57, 1425–1440, 1992.
E. Orlowska. Dynamic logic with program specifications and its relational proof system. Journal of Applied Non-Classical Logics, 3, 147–171, 1993.
E. Orlowska. Relational semantics for nonclassical logics: Formulas are relations. In: J. Wolenski, (ed), Philosophical Logic in Poland, pp. 167–186, Kluwer, Dordrecht, 1994.
V. R. Pratt. Semantical considerations on Floyd-Hoare logic. In: Proceedings of the 17th IEEE Symposium on Foundations of Computer Science, pp. 109–121, 1976.
H. Rasiowa. On generalized Post algebras of order w+ and w+- valued predicate calculi. Bulletin od the PAS, Ser. Math., 21, 209–219, 1973.
H. Rasiowa, R. Sikorski. The Mathematics of Metamathematics, Polish Science Publishers, Warsaw, 1963.
A. Tarski. On the calculus of relations. Journal of Symbolic Logic, 6, 73–89, 1941.
A. Tarski. Contributions to the theory of models. Indagationes Mathematicae, 17, 56–64, 1955.
D. Vakarelov. Modal logics for knoweldge representation systems. Theoretical Computer Science, 90, 433–456, 1991.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Orlowska, E. (1996). Relational Proof Systems for Modal Logics. In: Wansing, H. (eds) Proof Theory of Modal Logic. Applied Logic Series, vol 2. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2798-3_5
Download citation
DOI: https://doi.org/10.1007/978-94-017-2798-3_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4720-5
Online ISBN: 978-94-017-2798-3
eBook Packages: Springer Book Archive