Tableau Systems for Epistemic Positional Logics

Mateusz Klonowski; Krzysztof Aleksander Krawczyk; Bożena Pięta

doi:10.18778/0138-0680.2021.06

Authors

Mateusz Klonowski Nicolaus Copernicus University, Department of Logic, ul. Stanisława Moniuszki 16/20, 87-100 Torun, Poland Author https://orcid.org/0000-0001-8616-9189
Krzysztof Aleksander Krawczyk Nicolaus Copernicus University, Department of Logic, ul. Stanisława Moniuszki 16/20, 87-100 Torun, Poland Author https://orcid.org/0000-0003-4367-4796
Bożena Pięta Nicolaus Copernicus University, Department of Logic, ul. Stanisława Moniuszki 16/20, 87-100 Torun, Poland Author https://orcid.org/0000-0002-6281-0469

DOI:

https://doi.org/10.18778/0138-0680.2021.06

Keywords:

epistemic logic, logical omniscience, positional logic, tableau system

Abstract

The goal of the article is twofold. The first one is to provide logics based on positional semantics which will be suitable for the analysis of epistemic modalities such as ‘agent ... knows/beliefs that ...’. The second one is to define tableau systems
for such logics. Firstly, we present the minimal positional logic MR. Then, we change the notion of formulas and semantics in order to consider iterations of the operator of realization and “free” classical formulas. After that, we move on to weaker logics in order to avoid the well known problem of logical omniscience. At the same time, we keep the positional counterparts of modal axioms (T), (4) and (5). For all of the considered logics we present sound and complete tableau systems.

References

[1] T. Jarmużek, Minimal Logical Systems With R-operator: Their Metalogical Properties and Ways of Extensions, [in:] J. Bézieau, A. Costa-Leite (eds.), Perspectives on Universal Logic, Polimetrica Publisher (2007), pp. 319–333.

[2] T. Jarmużek, Tableau Metatheorem for Modal Logics, [in:] R. Ciuni, H. Wansing, C. Willkommen (eds.), Trends in Logic Vol. 41. Recent Trends in Philosophical Logic, Springer International Publishing (2014), pp. 103–126, DOI: https://doi.org/10.1007/978-3-319-06080-4_8 DOI: https://doi.org/10.1007/978-3-319-06080-4_8

[3] T. Jarmużek, On the Sea Battle Tomorrow That May Not Happen, Peter Lang, Berlin (2018), DOI: https://doi.org/10.3726/b14343 DOI: https://doi.org/10.3726/b14343

[4] T. Jarmużek, M. Klonowski, Some Intensional Logics Defined by Relating Semantics and Tableau Systems, [in:] A. Giordani, J. Malinowski (eds.), Logic in High Definition. Current Issues in Logical Semantics, Springer International Publishing (2021), pp. 31–48, DOI: https://doi.org/10.1007/978-3-030-53487-5_3 DOI: https://doi.org/10.1007/978-3-030-53487-5_3

[5] T. Jarmużek, A. Parol, On Some Language Extension of Logic MR: A Semantic and Tableau Approach, Roczniki Filozoficzne, vol. 68 (2021), pp. 345–366, DOI: https://doi.org/10.18290/rf20684-16 DOI: https://doi.org/10.18290/rf20684-16

[6] T. Jarmużek, A. Pietruszczak, Completeness of Minimal Positional Calculus, Logic and Logical Philosophy, vol. 13 (2004), pp. 147–162, DOI: https://doi.org/10.12775/LLP.2004.009 DOI: https://doi.org/10.12775/LLP.2004.009

[7] T. Jarmużek, M. Tkaczyk, Normalne logiki pozycyjne (Normal Positional Logics), Wydawnictwo KUL, Lublin (2015).

[8] T. Jarmużek, M. Tkaczyk, Expressive Power of the Positional Operator R: a Case Study in Modal Logic and Modal Philosophy, Ruch Filozoficzny, vol. 75 (2019), pp. 93–107, DOI: https://doi.org/10.12775/RF.2019.022 DOI: https://doi.org/10.12775/RF.2019.022

[9] A. Karczewska, Maximality of the Minimal R-Logic, Logic and Logical Philosophy, vol. 27 (2017), pp. 193–203, DOI: https://doi.org/10.12775/LLP.2017.008 DOI: https://doi.org/10.12775/LLP.2017.008

[10] J. Łos, Podstawy analizy metodologicznej kanonów Milla (The Foundations of Methodological Analysis of Mill’s Canons), Annales Universitatis Mariae Curie-Skłodowska, vol. 2 (1947), pp. 269–301.

[11] J. Łos, Logiki wielowartosciowe a formalizacja funkcji intensjonalnych (Many-Valued Logics and Formalization of Intensional Functions), Kwartalnik Filozoficzny, vol. 17(1) (1948), pp. 59–68.

[12] J. Malinowski, K. Pietrowicz, J. Szalacha-Jarmuzek, Logic of Social Ontology and Łos’s Operator, Logic and Logical Philosophy, vol. 29 (2020), pp. 239–258, DOI: https://doi.org/10.12775/LLP.2020.005 DOI: https://doi.org/10.12775/LLP.2020.005

[13] J. Meyer, Epistemic Logic, [in:] L. Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell (2001), pp. 183–202, DOI: https://doi.org/10.1002/9781405164801 DOI: https://doi.org/10.1111/b.9780631206934.2001.00012.x

[14] G. Priest, An Introduction to Non-Classical Logic. From If to Is. 2nd ed, Cambridge University Press, Cambridge (2008), DOI: https://doi.org/10.1017/CBO9780511801174 DOI: https://doi.org/10.1017/CBO9780511801174

[15] N. Rescher, On the Logic of Chronological Propositions, Mind, vol. 75(297) (1966), pp. 75–96, DOI: https://doi.org/10.2307/2251711 DOI: https://doi.org/10.1093/mind/LXXV.297.75

[16] N. Rescher, Topological Logic, [in:] Topics in Philosophical Logic, Reidel Publishing Company, Dordrecht (1968), pp. 229–249, DOI: https://doi.org/10.1007/978-94-017-3546-9_13 DOI: https://doi.org/10.1007/978-94-017-3546-9_13

[17] N. Rescher, A. Urquhart, Temporal Logic, Springer Verlag, Vienna–New York (1971). DOI: https://doi.org/10.1007/978-3-7091-7664-1

[18] M. Tkaczyk, Logika czasu empirycznego: funktor realizacji czasowej w językach teorii fizykalnych (Logic of Empirical Time: Functor of Temporal Realisation in the Languages of Phisical Theories), Wydawnictwo KUL, Lublin (2009).

[19] M. Tkaczyk, Negation in Weak Positional Calculi, Logic and Logical Philosophy, vol. 22 (2013), pp. 3–19, DOI: https://doi.org/10.12775/LLP.2013001 DOI: https://doi.org/10.12775/LLP.2013.001

[20] M. Tkaczyk, Distribution Laws in Weak Positional Logics, Roczniki Filozoficzne, vol. 66 (2018), pp. 163–179, DOI: https://doi.org/10.18290/rf.2018.66.3-8 DOI: https://doi.org/10.18290/rf.2018.66.3-8