Abstract
In this chapter we prove a functional completeness result for the hybrid logic of the universal modality. The chapter is structured as follows. In the first section of the chapter we describe the natural deduction system under consideration, in the second section we give an introduction to the notion of functional completeness, and in the third section we give general rule schemas for natural deduction rules. In the fourth section we prove the functional completeness result and in the final section we discuss the result.
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
Notes
- 1.
The fact that functional completeness results can be given for hybrid logics involving modalities interpreted using respectively the universal relation and the relation of inequality makes it tempting to draw a parallel to Tarski (1986) where Alfred Tarski argues that given an arbitrary set, there are only four binary relations on the set which should be called “logical”, namely the empty relation, the universal relation, the relation of equality, and the relation of inequality. Tarski’s argument is based on the observation that these relations are exactly the binary relations on the set which are mapped to themselves by all bijections on the set in question. Tarski motivated his argument by an analogy to Klein’s Erlangen Programm, named after the famous matematician Felix Klein (1849–1925). According to the Erlangen Programm, geometrical notions should be classified in terms of which transformations (bijective functions from a space to itself) they preserve. For example, all notions in Euclidean geometry are preserved by similarity transformations which are transformations that decrease or increase the size of a geometrical figure uniformly in all directions, hence, a triangle is transformed into a triangle with the same angles but possibly with proportionlly smaller or larger sides. It follows that notions which are not preserved by all similarity transformations, for example the notion of the distance between two pointe, cannot be formulated in Euclidean geometry. In an analogous way Tarski proposed to distinguish between logical and non-logical notions.
- 2.
Note that the symbol \(\boldsymbol\phi\) ranges over metavariables, thus, the symbol is a metametavariable. In general we let the metametavariables \(\boldsymbol\phi, {\boldsymbol\psi}, {\boldsymbol\theta}, \ldots\) range over the metavariables φ, ψ, θ, …. Observe that metametavariables are printed in boldface.
- 3.
In fact, if the language \({\cal H}\) is extended with a finite number \(\sharp_{1}, \ldots, \sharp_{s}\) of new connectives like the connective ♯ and the natural deduction system \(\textbf{N}_{{\cal H}}\) is extended with introduction and elimination rules for each of the new connectives, then all results of this chapter still holds provided no introduction rule for a connective ♯ i exhibits a connective ♯ j where \(j \neq i\). This would for example enable us to consider connectives with a varying number of inputs, like conjunction (but formally, the connectives \(\sharp_{1}, \ldots, \sharp_{s}\) are independent of each other even though they might be equipped with similar rules, like the rules for ternary conjunction are similar to the rules for binary conjunction).
- 4.
This prefixed tableau system is also considered in Fitting’s handbook chapter Fitting (2007) where it is shown to bear a direct relationship to a hypersequent system for S5, to be more precise, a hypersequent is a finite sequence of ordinary Gentzen sequents, and the sequents in a hypersequent play the role of the prefixes in a branch of a prefixed tableau. In Fitting (2007) this direct relationship is used to prove that completeness of the hypersequent system for S5 follows from completeness of the prefixed tableau system for S5. The direct relationship can be established since the prefix machinery for S5 is very simple as there is no accessibility relation in S5 models. It is not clear whether such a direct relationship can be established for other modal logics where the prefix machinery is more complicated.
References
M. de Rijke. The modal logic of inequality. Journal of Symbolic Logic, 57:566–584, 1992.
M. Fitting. Basic modal logic. In D.M. Gabbay et al., editors, Handbook of Logic in Artificial Intelligence and Logic Programming, volume 1, pages 365–448. Oxford University Press, Oxford, 1993.
G. Gargov and V. Goranko. Modal logic with names. Journal of Philosophical Logic, 22:607–636, 1993.
V. Goranko and S. Passy. Using the universal modality: Gains and questions. Journal of Logic and Computation, 2:5–30, 1992.
D.P. McCullough. Logical connectives for intuitionistic propositional logic. Journal of Symbolic Logic, 36:15–20, 1969.
M. Ohnishi and K. Matsumoto. Gentzen method in modal calculi. Osaka Mathematical Journal, 9:113–130, 1957.
M. Ohnishi and K. Matsumoto. Gentzen method in modal calculi, II. Osaka Mathematical Journal, 11:115–120, 1959.
A. Tarski. What are logical notions? History and Philosophy of Logic, 7:143–154, 1986. The paper is edited by J. Corcoran.
W. van der Hoek and M. de Rijke. Counting objects. Journal of Logic and Computation, 5: 325–345, 1995.
F. von Kutschera. Die Vollständigkeit des Operatorsystems \(\{ \neg, \wedge, \vee, \supset \} \) für die intuitionistische Aussagenlogik im Rahmen der Gentzensemantik. Archiv für Mathematische Logik und Grundlagenforschung, 11:3–16, 1968.
H. Wansing. The idea of a proof-theoretic semantics and the meaning of the logical operations. Studia Logica, 64:3–20, 2000.
H. Wansing. Logical connectives for constructive modal logic. Synthese, 150:459–482, 2006. Special issue edited by T. Braüner, P. Hasle, and P. Øhrstrøm.
J. Zucker and R.S. Tragesser. The adequacy problem for inferential logic. Journal of Philosophical Logic, 7:501–516, 1978.
C. Areces. Logic Engineering. The Case of Description and Hybrid Logics. PhD thesis, Institute for Logic, Language and Computation, University of Amsterdam, 2000.
M. Fitting. Modal proof theory. In P. Blackburn, J. van Benthem, and F. Wolter, editors, Handbook of Modal Logic, pages 85–138. Elsevier, 2007.
G. Mints. A Short Introduction to Modal Logic. CSLI Publications, 1992.
D. Prawitz. Natural Deduction. A Proof-Theoretical Study. Almqvist and Wiksell, Stockholm, 1965.
D. Prawitz. Proofs and the meaning and completeness of the logical constants. In J. Hintikka et al., editors, Essays on Mathematical and Philosophical Logic, Studies in Logic and The Foundations of Mathematics, pages 25–40. Reidel, 1978.
H. Wansing. A proof-theoretic proof of functional completeness for many modal and tense logics. In H. Wansing, editor, Proof Theory of Modal Logic, volume 2 of Applied Logic Series, pages 123–136. Kluwer, 1996.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Braüner, T. (2011). Functional Completeness for a Hybrid Logic. In: Hybrid Logic and its Proof-Theory. Applied Logic Series, vol 37. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0002-4_5
Download citation
DOI: https://doi.org/10.1007/978-94-007-0002-4_5
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-0001-7
Online ISBN: 978-94-007-0002-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)