David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Logica Universalis 3 (2):219-241 (2009)
In this paper we study an alternative approach to the concept of abstract logic and to connectives in abstract logics. The notion of abstract logic was introduced by Brown and Suszko (Diss Math 102:9–42, 1973)—nevertheless, similar concepts have been investigated by various authors. Considering abstract logics as intersection structures we extend several notions to their κ -versions ( κ ≥ ω ), introduce a hierarchy of κ -prime theories, which is important for our treatment of infinite connectives, and study different concepts of κ -compactness. We are particularly interested in non-topped intersection structures viewed as semi-lattices with a minimal meet-dense subset, i.e., with a minimal generator set. We study a chain condition which is sufficient for a minimal generator set, implies compactness of the logic, and in regular logics is equivalent to ( κ -) compactness of the consequence relation together with the existence of a ( κ -)inconsistent set, where κ is the cofinality of the cardinality of the logic. Some of these results are known in a similar form in the context of closure spaces, we give extensions to (non-topped) intersection structures and to big cardinals presenting new proofs based on set-theoretical tools. The existence of a minimal generator set is crucial for our way to define connectives. Although our method can be extended to further non-classical connectives we concentrate here on intuitionistic and infinite ones. Our approach leads us to the concept of the set of complete theories which is stable under all considered connectives and gives rise to the definition of the topological space of the logic. Topological representations of (non-classical) abstract logics by means of this space remain to be further investigated.
|Keywords||abstract logics intersection structures κ-compactness κ-prime theories connectives complete theories|
|Categories||categorize this paper)|
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
J. P. Cleave (1991). A Study of Logics. Oxford University Press.
J. Michael Dunn (2001). Algebraic Methods in Philosophical Logic. Oxford University Press.
Citations of this work BETA
Christian Wallmann (2013). A Shared Framework for Consequence Operations and Abstract Model Theory. Logica Universalis 7 (2):125-145.
Steffen Lewitzka (2011). ?K: A Non-Fregean Logic of Explicit Knowledge. Studia Logica 97 (2):233-264.
Similar books and articles
C. G. Fermüller (2008). Dialogue Games for Many-Valued Logics — an Overview. Studia Logica 90 (1):43 - 68.
Josep M. Font & Ventura Verdú (1989). A First Approach to Abstract Modal Logics. Journal of Symbolic Logic 54 (3):1042-1062.
Daniele Mundici (1986). Inverse Topological Systems and Compactness in Abstract Model Theory. Journal of Symbolic Logic 51 (3):785-794.
Marcelo Tsuji (1998). Many-Valued Logics and Suszko's Thesis Revisited. Studia Logica 60 (2):299-309.
Ramon Jansana (1995). Abstract Modal Logics. Studia Logica 55 (2):273 - 299.
Josep Maria Font & Miquel Rius (2000). An Abstract Algebraic Logic Approach to Tetravalent Modal Logics. Journal of Symbolic Logic 65 (2):481-518.
Petr Cintula (2005). Two Notions of Compactness in Gödel Logics. Studia Logica 81 (1):99 - 122.
M. Angela Weiss & Rohit Parikh (2002). Completeness of Certain Bimodal Logics for Subset Spaces. Studia Logica 71 (1):1 - 30.
M. E. Coniglio & W. A. Carnielli (2002). Transfers Between Logics and Their Applications. Studia Logica 72 (3):367-400.
Steffen Lewitzka (2007). Abstract Logics, Logic Maps, and Logic Homomorphisms. Logica Universalis 1 (2):243-276.
Added to index2009-08-17
Total downloads20 ( #130,548 of 1,699,719 )
Recent downloads (6 months)2 ( #269,935 of 1,699,719 )
How can I increase my downloads?