The Power of a Propositional Constant

Monomodal logic has exactly two maximally normal logics, which are also the only quasi-normal logics that are Post complete, and they are complete for validity in Kripke frames. Here we show that addition of a propositional constant to monomodal logic allows the construction of continuum many maximally normal logics that are not valid in any Kripke frame, or even in any complete modal algebra. We also construct continuum many quasi-normal Post complete logics that are not normal. The set of extensions of S4.3 is radically altered by the addition of a constant: we use it to construct continuum many such normal extensions of S4.3, and continuum many non-normal ones, none of which have the finite model property. But for logics with weakly transitive frames there are only eight maximally normal ones, of which five extend K4 and three extend S4.

[1]  Algebraic logic , 1985, Problem books in mathematics.

[2]  Krister Segerberg.  Post Completeness in Modal Logic , 1972, J. Symb. Log..

[3]  J. Heijenoort.  From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 , 1967 .

[4]  Robert S. Streett,et al.  Propositional Dynamic Logic of Looping and Converse Is Elementarily Decidable , 1982, Inf. Control..

[5]  H. P. Rickman,et al.  ESCAPISM: THE LOGICAL BASIS OF ETHICS , 1963 .

[6]  A. Chagrov,et al.  Modal Logic (Oxford Logic Guides, vol. 35) , 1997 .

[7]  P. Nowell-Smith,et al.  I.—ESCAPISM: THE LOGICAL BASIS OF ETHICS , 1960 .

[8]  Kit Fine,et al.  The Logics Containing S 4.3 , 1971 .

[9]  Robert Goldblatt,et al.  Monadic Bounded Algebras , 2010, Stud Logica.

[10]  Marcus Kracht,et al.  Semisimple Varieties of Modal Algebras , 2006, Stud Logica.

[11]  Günter Asser,et al.  Zeitschrift für mathematische Logik und Grundlagen der Mathematik , 1955 .

[12]  Tadeusz Litak,et al.  On Notions of Completeness Weaker than Kripke Completeness , 2004, Advances in Modal Logic.

[13]  Robert Goldblatt.  An algebraic study of well-foundedness , 1985, Stud Logica.

[14]  Krister Segerberg,et al.  An essay in classical modal logic , 1971 .

[15]  D. Gabbay,et al.  Handbook of Philosophical Logic, Volume II. Extensions of Classical Logic , 1986 .

[16]  Steven K. Thomason,et al.  Semantic analysis of tense logics , 1972, Journal of Symbolic Logic.

[17]  R. A. Bull.  That All Normal Extensions of S4.3 Have the Finite Model Property , 1966 .

[18]  Saul A. Kripke.  SEMANTICAL ANALYSIS OF MODAL LOGIC II. NON-NORMAL MODAL PROPOSITIONAL CALCULI , 2014 .

[19]  Timothy Williamson.  Continuum Many Maximal Consistent Normal Bimodal Logics with Inverses , 1998, Notre Dame J. Formal Log..

[20]  K. Fine.  Logics containing K4. Part II , 1985, Journal of Symbolic Logic.

[21]  Rohan French.  Denumerably Many Post-Complete Normal Modal Logics with Propositional Constants , 2012, Notre Dame J. Formal Log..

[22]  David Makinson,et al.  Some embedding theorems for modal logic , 1971, Notre Dame J. Formal Log..

[23]  Alan Ross Anderson,et al.  A REDUCTION OF DEONTIC LOGIC TO ALETHIC MODAL LOGIC , 1958 .

[24]  Guram Bezhanishvili,et al.  An Algebraic Approach to Subframe Logics. Modal Case , 2011, Notre Dame J. Formal Log..

[25]  Kit Fine.  Logics Containing K4. Part I , 1974, J. Symb. Log..

[26]  K. Segerberg.  That Every Extension Of S4.3 is Normal , 1975 .

[27]  Leo Esakia,et al.  Intuitionistic logic and modality via topology , 2004, Ann. Pure Appl. Log..

[28]  Saul A. Kripke,et al.  Semantical Analysis of Modal Logic I Normal Modal Propositional Calculi , 1963 .

[29]  P. Soden.  University of Washington Press , 2008 .

[30]  Tomasz Kowalski,et al.  Varieties of Tense Algebras , 1998, Reports Math. Log..

[31]  Bernhard Beckert,et al.  Dynamic Logic , 2007, The KeY Approach.

[32]  Michael Zakharyaschev,et al.  Modal Logic , 1997, Oxford logic guides.

[33]  A. Tarski,et al.  Boolean Algebras with Operators. Part I , 1951 .

[34]  Kit Fine,et al.  Logics containing K4. Part I , 1974, Journal of Symbolic Logic.

[35]  J. Ferreirós.  From Frege to Gödel. A Source Book in Mathematical Logic, 1879¿1931: By Jean van Heijenoort. Cambridge, MA (Harvard University Press). 1967; new paperback edn., 2002. 664 pages, 1 halftone. ISBN: 0-674-32449-8. $27.95 , 2004 .

[36]  Emil L. Post.  Introduction to a General Theory of Elementary Propositions , 1921 .