Abstract
The aim of the paper is to demonstrate and prove a tableau metatheorem for modal logics. While being effective tableau methods are usually presented in a rather intuitive way and our ambition was to expose the method as rigorously as possible. To this end all notions displayed in the sequel are couched in a set theoretical framework, for example: branches are sequences of sets and tableaus are sets of these sequences. Other notions are also defined in a similar, formal way: maximal, open and closed branches, open and closed tableaus. One of the distinctive features of the paper is introduction of what seems to be the novelty in the literature: the notion of tableau consequence relation. Thanks to the precision of tableau metatheory we can prove the following theorem: completeness and soundness of tableau methods are immediate consequences of some conditions put upon a class of models M and a set of tableau rules MRT. These conditions will be described and explained in the sequel. The approach presented in the paper is very general and may be applied to other systems of logic as long as tableau rules are defined in the style proposed by the author. In this paper tableau tools are treated as an entirely syntactical method of checking correctness of arguments [1, 2].
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Notes
- 1.
The author would like to thank an anonymous reviewer for many valuable comments and suggestions that have allowed to improve the paper.
- 2.
In the literature this kind of model is usually called a pointed model, but we will shortly call it model.
- 3.
We use a word inconsistent instead of—for example—contradictory, since it enables us to do a direct transition between semantic and tableau notions.
- 4.
When we impose the condition of t-consistency, it seems we are not able to capture the sound rule \(\{A, \lnot A \} \models B\). This is not the case, because—as we will see—starting from a set of tableau premisses \(\{\langle A, i\rangle , \langle \lnot A, i \rangle , \langle \lnot B, i \rangle \}\), for some formulas \(A, B \in \mathsf{For }\) and some index \(i \in {\mathbb {N}}\), immediately we have a closed tableau, so the rule is sound.
- 5.
It does not mean that the set of all instances of any rule in \(\mathbf R \) is finite.
- 6.
This condition is not redundant. In the general definition of a tableau 1.28 every branch must be \(\varPhi \)-maximal (according to Definition 1.27), here must be just maximal.
- 7.
See Chap. 4.4 of [3].
References
Jarmużek, T. (2008). Tableau system for logic of categorial propositions and decidability. Bulletin of the Section of Logic, 37(3/4), 223–231.
Jarmużek, T. (2013). Formalizacja metod tablicowych dla logik zdań i logik nazw ( Formalization of tableau methods for propositional logics and for logics of names). Toruń: Wydawnictwo UMK.
Blackburn, P., de Rijke, M., & Vennema, Y. (2002). Modal logic. Cambridge.
Rajeev, G. Tableau methods for modal and temporal logics (pp. 297—396) (in [1]).
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Jarmużek, T. (2014). Tableau Metatheorem for Modal Logics. In: Ciuni, R., Wansing, H., Willkommen, C. (eds) Recent Trends in Philosophical Logic. Trends in Logic, vol 41. Springer, Cham. https://doi.org/10.1007/978-3-319-06080-4_8
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