On Some Language Extension of Logic MR: A Semantic and Tableau Approach

Keywords: extension of minimal positional logic, MR, positional logic, realization operator, tableau methods


In the article we present an extension of the minimal, normal positional logic, i.e., the logic with realization operator MR. Positional logic is a philosophical logic that makes it possible to relate sentences to contexts that can be understood in many ways. We enrich the basic language of minimal positional logic with additional expressions built with predicates and positional constants. We also accept expressions built with the realization operator and many positions, like:


Thanks to this, we increased the expressivity of minimal positional logic. In the article we point to many examples of the fact that, thanks to this small change, complex theories based on the proposed extension can be created. As a theory of proof for our logic, we assume tableau methods, showing soundness and completeness theorems. At the end, however, we show that the logic studied here is only a language extension of the MR: all theorems of the extension have their equivalents in pure MR theorems. However, theories built upon the proposed extension can express much more than theories built upon pure MR.


Hiż, Henryk. 1951. Review: “Jerzy Los. Foundations of the Methodological Analysis of Mill’s Canons.” Journal of Symbolic Logic 16, issue 1: 58–59.

Jarmużek, Tomasz, and Andrzej Pietruszczak. 2004. “Completeness of Minimal Positional Calculus.” Logic and Logical Philosophy 13: 147–162. doi: 10.12775/ LLP.2018.013.

Jarmużek, Tomasz, and Marcin Tkaczyk. 2015. Normalne logiki pozycyjne [Normal Positional Logics]. Lublin: Towarzystwo Naukowe KUL.

Jarmużek, Tomasz, and Marcin Tkaczyk. 2019. “Jerzy Łoś Positional Calculus and the Origin of Temporal Logic.” Logic and Logical Philosophy 28: 259–276. doi: 10.12775/LLP. 2018.013

Jarmużek, Tomasz. 2013. Jutrzejsza bitwa morska. Rozumowanie Diodora Kronosa [Tomorrow Sea–Fight: Diodorus Cronus’ Argument]. Toruń, Wydawnictwo Naukowe UMK.

Jarmużek, Tomasz. 2007. “Minimal Logical Systems with -operator: Their Metalogical Properties and Ways of Extensions.” In Perspectives on Universal Logic, edited by Jean-Yves Béziau and Alexandre Costa-Leite, 319–33. Polimetrica International Scientific Publisher, Italy.

Jarmużek, Tomasz. 2018. On the Sea Battle Tomorrow That May Not Happen: A Logical and Philosophical Analysis of the Master Argument. Berlin, Warsaw: Peter Lang Publishing.

Jarmużek, Tomasz. 2006. “Rekonstrukcje Rozumowania Diodora Kronosa w ontologii czasu punktowego [Reconstruction of Diodorus Cronus’ Argument in Frame of Ontology of Time Consisted of Points].” Analiza i Egzystencja 3: 197–215.

Jarmużek, Tomasz, and Mateusz Klonowski. 2020. “Some Intensional Logics Defined by Relating Semantics and Tableau Systems.” In Logic in High Definition. Trends in Logical Semantics, edited by Alessandro Giordani and Jacek Malinowski, 33–51. Wien, New York: Springer Verlag

Łoś, Jerzy. 1948. “Logiki wielowartościowe a formalizacja funkcji intensjonalnych [Multivalued Logics and Formalization of Intensional Functions].” Kwartalnik Filozoficzny 17, No. 1-2: 59–78.

Łoś, Jerzy. 1947. “Podstawy analizy metodologicznej kanonów Milla [Foundations of Methodological Analysis of Mill’s Canons].” Annales Universitatis Mariae Curie-Skłodowska 2.5. F: 269–301.

Karczewska, Anna Maria. 2018. “Maximality of the Minimal -logic.” Logic and Logical Philosophy 27: 193–203. doi: 10.12775/LLP.2017.008

Lechniak, Marek. 1988. “Logika epistemiczna Jerzego Łosia a teoria racjonalnego zachowania [Epistemic Logic of Jerzy Łoś and the Theory of Rational Behaviour].” Roczniki Filozoficzne 26, issue 1: 79–91.

Malinowski, Jacek, Krzysztof Pietrowicz, and Joanna Szalacha-Jarmużek. 2020. “Logic of Social Ontology and Łoś Operator.” Logic and Logical Philosophy 29: 239–258. doi: 10.12775/LLP.2020.005.

Rasiowa, Helena, and Roman Sikorski. 1968. Mathematics of Metamathematics. 2nd editon. Warszawa: PWN.

Rescher, Nicholas, and Alasdair Urquhart. 1971. Temporal Logic, Library of Exact Philosophy. Wien, New York: Springer Verlag.

Garson, James, and Nocholas Rescher. 1968. “Topological Logic.” Journal of Symbolic Logic 33, issue 4: 537–48.

Suszko, Roman. 1949. Review: “Jerzy Los, Many-Valued Logics and Formalization of Intensional Functions.” Journal of Symbolic Logic 14: 64–65.

Tkaczyk, Marcin. 2009. Logika czasu empirycznego [Logic of Physical Time]. Lublin: Wydawnictwo KUL.

Tkaczyk, Marcin. 2013. “Distribution Laws in Weak Positional Logics.” Roczniki Filozoficzne 66, nr 3: 163–179. doi: 10.18290/rf.2018.66.3-8.

Tkaczyk, Marcin. 2013. “Negation in Weak Positional Calculi.” Logic and Logical Philosophy 22: 3–19. doi: 10.12775/LLP.2013.001.