Logic Journal of IGPL Advance Access originally published online on February 26, 2008
Logic Journal of IGPL 2008 16(3):233-248; doi:10.1093/jigpal/jzn002
Categorical Abstract Algebraic Logic: Bloom's Theorem for Rule-Based 
-Institutions
School of Mathematics and Computer Science, Lake Superior State University, Sault Sainte Marie, MI 49783, USA.
E-mail: gvoutsad{at}lssu.edu
A syntactic machinery is developed for 
-institutions based on the notion of a category of natural transformations on their sentence functors. Rules of inference, similar to the ones traditionally used in the sentential logic framework to define the best known sentential logics, are, then, introduced for -institutions. A -institution is said to be rule-based if its closure system is induced by a collection of rules of inference. A logical matrix-like semantics is introduced for rule-based -institutions and a version of Bloom's Lemma and Bloom's Theorem are proved for rule-based -institutions.
Key Words: Closure Operators • Deductive Systems • Logical Matrices • Universal Horn Logic Without Equality • Bloom's Theorem • -Institutions • Rules of Inference • Filtered Products • Ultraproducts • 2000 AMS Subject Classification: • Primary: 03G99 • 18C15 • Secondary: 68N30
Received for publication .
References
-
1 Andréka H, Németi I. Studia Scientiarum Mathematicarum Hungarica.
o
Lemma Holds in Every Category (1978) Vol. 13:361–376.2 Andréka H, Németi I. Colloquia Mathematics Societas János Bolyai. A General Axiomatizability Theorem Formulated in Terms of Cone-Injective Subcategories (1981) Vol. 29:13–35.
3 Barr M, Wells C. Category Theory for Computing Science. (1999) Third Edition. Montréal: Les Publications CRM.
4 Blok WJ, Pigozzi D. Studia Logica. Protoalgebraic Logics (1986) Vol. 45:337–369.
5 Blok WJ, Pigozzi D. Memoirs of the American Mathematical Society. Algebraizable Logics (1989) Vol. 77(No. 396).
6 Blok WJ, Pigozzi D. Algebraic Semantics for Universal Horn Logic With-out Equality, in Universal Algebra and Quasigroup Theory. Romanowska A, Smith J.D.H, eds. (1992) Berlin: Heldermann Verlag.
7 Bloom SL. Studia Logica. Some Theorems on Structural Consequence Operations (1975) Vol. 34:1–9.
8 Borceux F. Handbook of Categorical Algebra, Encyclopedia of Mathematics and its Applications. (1994) Vol. 50. Cambridge, U.K: Cambridge University Press.
9 Chang CC, Keisler HJ. Model Theory. (1973) Amsterdam: North-Holland.
10 Czelakowski J. Studia Logica. Equivalential Logics I,II (1981) Vol. 40:227–236. 355–372.
11 Czelakowski J. Studia Logica Library 10. In: Protoalgebraic Logics (2001) Dordrecht: Kluwer.
12 Diaconescu R. Fundamenta Informaticae. Institution-Independent Ultraproducts (2003) Vol. 55:321–348.
13 Doets K. Basic Model Theory. (1996) Stanford: CSLI Publications and FoLLI.
14 Fiadeiro J, Sernadas A. Structuring Theories on Consequence, in Recent Trends in Data Type Specification. Lecture Notes in Computer Science—Sannella Donald, Tarlecki Andrzej, eds. (1988) Vol. 332. New York: Springer-Verlag. 44–72.[Web of Science]
15 Font JM, Jansana R. Lecture Notes in Logic. In: A General Algebraic Semantics for Sentential Logics (1996) Vol. 7. Berlin Heidelberg: Springer-Verlag.
16 Font JM, Jansana R, Pigozzi D. Studia Logica. A Survey of Abstract Algebraic Logic (2003) Vol. 74(No. 1/2):13–97.
17 Goguen JA, Burstall RM. Proceedings of the Logic of Programming Workshop, Lecture Notes in Computer Science. In: Introducing Institutions—Clarke E, Kozen D, eds. (1984) Vol. 164. New York: Springer-Verlag. 221–256.
18 Goguen JA, Burstall RM. Journal of the Association for Computing Machinery. Institutions: Abstract Model Theory for Specification and Programming (1992) Vol. 39(No. 1):95–146.
19 Herrmann B. Equivalential Logics and Definability of Truth. (1993) Berlin: Dissertation, Freie Universitat Berlin.
20 Herrmann B. Studia Logica. Equivalential and Algebraizable Logics (1996) Vol. 57:419–436.
21 Herrmann B. Studia Logica. Characterizing Equivalential and Algebraizable Logics by the Leibniz Operator (1997) Vol. 58:305–323.
22 Hodges W. Model Theory. (1993) Cambridge: Cambridge University Press.
23 o J. Studia Logica. An Algebraic Treatment of the Methodology of Elementary Deductive Systems (1955) Vol. 2:151–212.
24 Mac Lane S. Categories for the Working Mathematician. (1971) Springer-Verlag.
25 Prucnal T, Wro
ski A. Bulletin of the Section of Logic. An algebraic characterization of the notion of structural completeness (1974) Vol. 3:30–33.
26 Tarlecki A. Lecture Notes in Computer Science. Bits and Pieces of the Theory of Institutions (1986) Vol. 240:334–360.
27 Tarski A. Über einige fundamentale Begriffe der Metamathematik (1930) Vol. 23:22–29. C.R. Soc. Sci. Lettr. Varsovie, Cl. III.
28 Voutsadakis G. Submitted to the Annals of Pure and Applied Logic. In: Categorical Abstract Algebraic Logic: Tarski Congruence Systems, Logical Morphisms and Logical Quotients. Preprint available at http://voutsadakis.com/RESEARCH/papers.html.
29 Voutsadakis G. Notre Dame Journal of Formal Logic. Categorical Abstract Algebraic Logic: Models of -Institutions (2005) Vol. 46(No. 4):439–460.
30 Voutsadakis G. Categorical Abstract Algebraic Logic: Generalized Tarski Congruence Systems. Preprint available at http://voutsadakis.com/RESEARCH/papers.html.
31 Voutsadakis G. Applied Categorical Structures. Categorical Abstract Algebraic Logic: (I,N)-Algebraic Systems (2005) Vol. 13(No. 3):265–280.
32 Voutsadakis G. Scientiae Mathematicae Japonicae. Categorical Abstract Algebraic Logic: Gentzen -Institutions (e-2007) 407–422.
33 Voutsadakis G. Reports on Mathematical Logic. Categorical Abstract Algebraic Logic: Full Models, Frege Systems and Metalogical properties (2006) Vol. 41:31–62.
34 Voutsadakis G. Studia Logica, Studia Logica. Categorical Abstract Algebraic Logic: Prealgebraicity and Protoalgebraicity (2007) Vol. 85(No. 2):217–251.
35 Voutsadakis G. Notre Dame Journal of Formal Logic. Categorical Abstract Algebraic Logic: More on Protoalgebraicity (2006) Vol. 47(No. 4):487–514.
36 Voutsadakis G. Scientiae Mathematicae Japonicae. Categorical Abstract Algebraic Logic: Protoalgebraicity and Leibniz Theory Systems (2005) Vol. 62(No. 1):109–117.
37 Voutsadakis G. Mathematical Logic Quarterly. Categorical Abstract Algebraic Logic: The Largest Theory System Included in a Theory Family (2006) Vol. 52(No. 3):288–294.
38 Voutsadakis G. Submitted to the Archive for Mathematical Logic. In: Categorical Abstract Algebraic Logic: Operations on Classes of Models. Preprint available at http://voutsadakis.com/RESEARCH/papers.html.
39 Voutsadakis G. Applied Categorical Structures. Categorical Abstract Algebraic Logic: Partially Ordered Algebraic Systems (2006) Vol. 14(No. 1):81–98.
40 Voutsadakis G. Submitted to Algebra Universalis. In: Categorical Abstract Algebraic Logic: Closure Operators on Classes of PoFunctors. Preprint available at http://voutsadakis.com/RESEARCH/papers.html.
CiteULike 
Connotea 
Del.icio.us What's this?
| ||||||||||||||||||||||||||||||||||||||||||||||||