Archive for Mathematical Logic 50 (1-2):45-73 (2011)

We continue the work of Blok and Jónsson by developing the theory of structural closure operators and introducing the notion of a representation between them. Similarities and equivalences of Blok-Jónsson turn out to be bijective representations and bijective structural representations, respectively. We obtain a characterization for representations induced by a transformer. In order to obtain a similar characterization for structural representations we introduce the notions of a graduation and a graded variable of an M-set. We show that several deductive systems, Gentzen systems among them, are graded M-sets having graded variables, and describe the graded variables in each case. In the last section we show that, for a sentential logic, having an algebraic semantics is equivalent to being representable in an equational consequence. This motivates the extension of the notion of having an algebraic semantics for Gentzen systems, hypersequents systems, etc. We prove that if a closure operator is representable by a transformer, then every extension of it is also representable by the same transformer. As a consequence we obtain that if one of these systems has an algebraic semantics, then so does any of its extensions with the same defining equations
Keywords Structural closure operators  Representations  Transformers  Graduations  Graded variables  Algebraizability
Categories (categorize this paper)
DOI 10.1007/s00153-010-0201-z
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

PhilArchive copy

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 52,792
External links

Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library

References found in this work BETA

A Survey of Abstract Algebraic Logic.J. M. Font, R. Jansana & D. Pigozzi - 2003 - Studia Logica 74 (1-2):13 - 97.
Equivalence of Consequence Operations.W. J. Blok & Bjarni Jónsson - 2006 - Studia Logica 83 (1-3):91-110.
Weakly Algebraizable Logics.Janusz Czelakowski & Ramon Jansana - 2000 - Journal of Symbolic Logic 65 (2):641-668.
Correspondences Between Gentzen and Hilbert Systems.J. G. Raftery - 2006 - Journal of Symbolic Logic 71 (3):903 - 957.

View all 7 references / Add more references

Citations of this work BETA

The Semantic Isomorphism Theorem in Abstract Algebraic Logic.Tommaso Moraschini - 2016 - Annals of Pure and Applied Logic 167 (12):1298-1331.

Add more citations

Similar books and articles

Representations Gone Mental.Alex Morgan - 2014 - Synthese 191 (2):213-244.
The Classification of Small Types of Rank Ω, Part I.Steven Buechler & Colleen Hoover - 2001 - Journal of Symbolic Logic 66 (4):1884-1898.
Epistemic Modals.Seth Yalcin - 2007 - Mind 116 (464):983-1026.
Ramseyfication and Theoretical Content.Joseph Melia & Juha Saatsi - 2006 - British Journal for the Philosophy of Science 57 (3):561-585.
Structural Representations and the Brain.Oron Shagrir - 2012 - British Journal for the Philosophy of Science 63 (3):519-545.
On Closure in Economics.Stephen John Nash - 2004 - Journal of Economic Methodology 11 (1):75-89.


Added to PP index

Total views
79 ( #119,041 of 2,342,397 )

Recent downloads (6 months)
1 ( #514,928 of 2,342,397 )

How can I increase my downloads?


My notes