Abstract
The aim of this paper is to show how to simply define paraconsistent tableau systems by liberalization of construction of complete tableaus. The presented notions allow us to list all tableau inconsistencies that appear in a complete tableau. Then we can easily choose these inconsistencies that are effects of interactions between premises and a conclusion, simultaneously excluding other inconsistencies. A general technique we describe is presented here for the case of Propositional Logic, as the simplest one, but it can be easily extended to more complex cases. In other words, a kind of paraconsistent consequence relation is being studied here, and a simple tableau system is shown to exist that captures that consequence relation.
This work was completed with the support of Polish National Center of Science NCN UMO-2012/05/E/HS1/03542.
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Notes
- 1.
We use a word inconsistent instead—for example—contradictory, since it enables us a direct transition between semantical and tableau notions.
- 2.
We mean such logics that are logics of terms or propositions, and are two–valued.
- 3.
The rules modified a little in a manner that is good for our paraconsistent aim are presented in the Sect. 14.3.
- 4.
Generally, we divide complete branches into open and closed ones, since in our formal theory of tableau methods in [2] our aim is always to complete a branch, so a branch itself is just a technical concept. At the same time an occurrence of a t-inconsistency completes a branch. In the paper we change our point of view a bit: applying of rules is allowed as far as it is possible, ignoring any t-inconsistency—later we will come back to the idea, when explaining exactly what we mean by ‘blind rules’ (exactly in the Sect. 14.3).
References
Priest, G.: An Introduction to Non-Classical Logic. Cambridge University Press, Cambridge (2001)
Jarmużek, T.: Formalizacja metod tablicowych dla logik zdań i logik nazw (Formalization of tableau methods for propositional logics and for logics of names). Wydawnictwo UMK, Toruń (2013)
Jarmużek, T.: Tableau Metatheorem for Modal Logics. In: Ciuni, R., Wansing, H., Willkommen, C. (eds.) Recent Trends in Philosophical Logic, pp. 105–123. Springer (2013)
Acknowledgments
Many thanks to an anonymous reviewer for helpful comments and suggestions.
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Jarmużek, T., Tkaczyk, M. (2015). A Method of Defining Paraconsistent Tableaus. In: Beziau, JY., Chakraborty, M., Dutta, S. (eds) New Directions in Paraconsistent Logic. Springer Proceedings in Mathematics & Statistics, vol 152. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2719-9_14
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DOI: https://doi.org/10.1007/978-81-322-2719-9_14
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