Modal Propositional Truth Logic Tr and its Completeness


A. S. Karpenko
A. V. Chagrov


In this paper Sobochin_ski’s four-valued modal logic $\textbf{V2}$ (extension of $ \textbf{S5} $) is considered. The emergence of that logic, some its interesting properties and different equivalent formulations are presented. Its algebraic models are of particular interest: as the extension of De Morgan algebra by boolean negation $\neg$ and as the extension of Boolean algebra by the endomorphism $\textit{g}$, which is interpreted then as the propositional truth operation $T$. The logic corresponding to the last case is denoted by $Tr$. The attention is paid to the application of $Tr$ in Fitting’s theory of truth. The axiomatization of $Tr$ in language ($\rightarrow, \neg, T$) is considered. The completeness of logic $Tr$ is proved with use of Sahlqvist’s powerful theorem, which gives the sufficient condition of Kripke completeness for normal modal logics. Algebraic completeness of logic $Tr$ is also proved.






Ермолаева Н.М., Мучник А.А. Модальные расширения логических исчислений типа Хао Вана // Исследования по формализованным языкам и неклассическим логикам. М.: Наука, 1974. С. 172–193.
Ермолаева Н.М., Мучник А.А. Модальные логики, определяемые эндоморфизмами дистрибутивных решеток // Исследования по теории множеств и неклассическим логикам. М.: Наука, 1976. С. 229–246.
Карпенко А.С. Решетки четырехзначных модальных логик // Логические исследования. 2015. № 21(1). C. 122–137.
Клини С.К. Введение в метаматематику. М.: Иностранная литература, 1957. 527 с.
Леммон Е. Алгебраическая семантика для модальных логик I // Семантика модальных и интенсиональных логик / Ред. В.А. Смирнов. М.: Прогресс, 1981. С. 98–124.
Максимова Л. Л. Интерполяционные теоремы в модальных логиках и амальгамируемые многообразия топобулевых алгебр // Алгебра и логика. 1979. T. 18(5). С. 556–586.
Chagrov A., Zakharyaschev M. Modal Logic. Oxford: Clarendon Press, 1997. 624 p.
Fitting M. Bilattices and the theory of truth // Journal of Philosophical Logic. 1989. Vol. 18. P. 225–256.
Font J.M. Belnap’s four-valued logic and De Morgan lattices // Logic Journal of the IGPL. 1997. Vol. 5(3). P. 413–440.
Ginsberg M.L. Multivalued logics: A uniform approach to inference in artificial intelligence // Computational Intelligence. 1988. Vol. 4(3). P. 265–315.
Kripke S. Outline of a theory of truth // Journal of Philosophy. 1975. Vol. 72. P. 690–716.
Lewis C.I., Langford C.H. Symbolic Logic. N.Y.: Dover Publications, 1959 (2nd ed. with corrections). 506 p.
Pynko A.P. Functional completeness and axiomatizability within Belnap’s four-valued logic and its expansion // Journal of Applied Non-Classical Logics. 1999. Vol. 9(1). P. 61–105.
Sahlqvist H. Completeness and correspondence in the first and second order semantics for modal logic / Ed. S. Kanger. Proceedings of the Third Scandinavian Logic symposium. Amsterdam: North-Holland, 1975. P. 110–143.
Scroggs S.J. Extensions of the Lewis system S5 // The Journal of Symbolic Logic. 1951. Vol. 16. P. 112–120.
Sobochinski B. Modal system S4.4 // Notre Dame Journal of Formal Logic. 1964. Vol. 5(4). P. 305–312.
Sobochinski B. Certain extensions of modal system S4 // Notre Dame Journal of Formal Logic. 1970. Vol. 11(3). P. 347–367.