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Logic Journal of IGPL Advance Access published online on February 26, 2008
Logic Journal of IGPL, doi:10.1093/jigpal/jzn002
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© The Author, 2008. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Categorical Abstract Algebraic Logic: Bloom's Theorem for Rule-Based {pi}-Institutions

George Voutsadakis

School of Mathematics and Computer Science, Lake Superior State University, Sault Sainte Marie, MI 49783, USA.

E-mail: gvoutsad{at}lssu.edu


  Abstract

A syntactic machinery is developed for {pi}-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 {pi}-institutions. A {pi}-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 {pi}-institutions and a version of Bloom's Lemma and Bloom's Theorem are proved for rule-based {pi}-institutions.

Key Words: Closure Operators • Deductive Systems • Logical Matrices • Universal Horn Logic Without Equality • Bloom's Theorem • {pi}-Institutions • Rules of Inference • Filtered Products • Ultraproducts • 2000 AMS Subject Classification: • Primary: 03G99 • 18C15 • Secondary: 68N30


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