Completeness of S4 for the Lebesgue Measure Algebra
Journal of Philosophical Logic 41 (2):287-316 (2012)
| Abstract | We prove completeness of the propositional modal logic S 4 for the measure algebra based on the Lebesgue-measurable subsets of the unit interval, [0, 1]. In recent talks, Dana Scott introduced a new measure-based semantics for the standard propositional modal language with Boolean connectives and necessity and possibility operators, and . Propositional modal formulae are assigned to Lebesgue-measurable subsets of the real interval [0, 1], modulo sets of measure zero. Equivalence classes of Lebesgue-measurable subsets form a measure algebra, , and we add to this a non-trivial interior operator constructed from the frame of ‘open’ elements—elements in with an open representative. We prove completeness of the modal logic S 4 for the algebra . A corollary to the main result is that non-theorems of S 4 can be falsified at each point in a subset of the real interval [0, 1] of measure arbitrarily close to 1. A second corollary is that Intuitionistic propositional logic (IPC) is complete for the frame of open elements in | |||||||||
| Keywords | No keywords specified (fix it) | |||||||||
| Categories | ||||||||||
| Options |
|
|||||||||
Download options| PhilPapers Archive |
Upload a copy of this paper Check publisher's policy on self-archival Papers currently archived: 5,631 |
| External links |
 |
| Through your library | Configure |
Similar books and articlesMaxim R. Burke (1991). Powers of the Ideal of Lebesgue Measure Zero Sets. Journal of Symbolic Logic 56 (1):103-107.
Haim Judah & Saharon Shelah (1990). The Kunen-Miller Chart (Lebesgue Measure, the Baire Property, Laver Reals and Preservation Theorems for Forcing). Journal of Symbolic Logic 55 (3):909-927.
Stephen G. Simpson (2009). Mass Problems and Measure-Theoretic Regularity. Bulletin of Symbolic Logic 15 (4):385-409.
Peter Cholak, Noam Greenberg & Joseph S. Miller (2006). Uniform Almost Everywhere Domination. Journal of Symbolic Logic 71 (3):1057 - 1072.
I. A. Kieseppä (1996). Truthlikeness for Hypotheses Expressed in Terms of N Quantitative Variables. Journal of Philosophical Logic 25 (2):109 - 134.
Giovanni Sambin (1999). Subdirectly Irreducible Modal Algebras and Initial Frames. Studia Logica 62 (2):269-282.
Ernst Zimmermann (2003). Elementary Definability and Completeness in General and Positive Modal Logic. Journal of Logic, Language and Information 12 (1):99-117.
Jacek Cichoń & Janusz Pawlikowski (1986). On Ideals of Subsets of the Plane and on Cohen Reals. Journal of Symbolic Logic 51 (3):560-569.
Aldo Ursini (1979). A Modal Calculus Analogous to K4w, Based on Intuitionistic Propositional Logic, Iℴ. Studia Logica 38 (3):297 - 311.
Milan Božić & Kosta Došen (1984). Models for Normal Intuitionistic Modal Logics. Studia Logica 43 (3):217 - 245.
Jaime I. Ihoda & Saharon Shelah (1989). Martin's Axioms, Measurability and Equiconsistency Results. Journal of Symbolic Logic 54 (1):78-94.
Marion Scheepers (1999). Finite Powers of Strong Measure Zero Sets. Journal of Symbolic Logic 64 (3):1295-1306.
Anna Bucalo (1994). Modalities in Linear Logic Weaker Than the Exponential “of Course”: Algebraic and Relational Semantics. Journal of Logic, Language and Information 3 (3):211-232.
Analytics
Monthly downloads |
Added to index2012-03-14Total downloads9 ( #113,850 of 548,939 )Recent downloads (6 months)1 ( #63,492 of 548,939 )How can I increase my downloads? |
My notes
Discussion
